O wzorach służących do obliczenia liczby liczb pierwszych nie przekraczających danej granicy

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Autor	Antoni Baranowski
Tytuł	O wzorach służących do obliczenia liczby liczb pierwszych nie przekraczających danej granicy
Data wyd.	1895
Źródło	skany na Commons
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Indeks stron

[3]

Ks. Biskup A. Baranowski.

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O WZORACH

SŁUŻĄCYCH DO

OBLICZENIA LICZBY LICZB PIERWSZYCH

NIE PRZEKRACZAJĄCYCH DANEJ GRANICY.

W KRAKOWIE.

NAKŁADEM AKADEMII UMIEJĘTNOŚCI.

SKŁAD GŁÓWNY W KSIĘGARNI SPÓŁKI WYDAWNICZEJ POLSKIEJ.

1895.

[4]

Osobne odbicie z Tomu XXVIII. Rozpraw Wydziału matematyczno-przyrodniczego
Akademii Umiejętności w Krakowie.

Kraków, 1895. — W drukarni Uniwersytetu Jagiellońskiego, pod zarządem A. M. Kosterkiewicza.

[5]

O wzorach

służących do obliczenia liczby liczb pierwszych

nie przekraczających danej granicy.

Przez

ks. biskupa A. Baranowskiego.

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Rzecz przedstawiona na posiedzeniu Wydz. mat.-przyr. z dnia 2. lipca 1894 r.;

referent czł. Mertens.

 Od lat kilku zupełnie zależałem pole gramatyki, a ugrzązłem, raczej pogrążyłem się w matematyce, nie jako fachowy, ale jak dyletant, samouczek, bez prawideł, wskazówek i znajomości powszechnie przyjętych formuł. Napisałem kwadraty i sześciany liczb od $1$ do $1000$ z ich różnicami. Potęgowałem liczby małe: $2,$ $2^{2},$ $2^{3}\ldots$ $2^{300};$ $3,$ $3^{2},$ $3^{3}\ldots$ $3^{200};$ $5,$ $5^{2},$ $5^{3}\ldots$ $5^{150};$ $7,$ $7^{2},$ $7^{3}\ldots$ $7^{85}.$ Przy tej pracy poznałem stosunki i prawidła, o których później się dowiedziałem, że nazywane są dwumianem Newton’a. Zastanawiając się nad stosunkami i prawami wykładników, wpadłem na trop, który mię doprowadził do zbadania wielokształtu podzielności liczb złożonych. Przytem zbadałem wszystkie liczby w zakresie numeracyi od $0$ do $1000$; potem do $10\,000$. Za wskazówką i poradą Dra Webera zapisałem sobie dzieło Wertheima „Die Theorie der Zahlen“, ale jego wykład i znakowanie — dla mnie, jako laika, były orzechami zanadto twardemi na moje zęby. Zainteresowała mię jednak funkcya $\operatorname {\varphi } (m)$, jako mająca związek z moją analizą liczb w celu znalezienia prawideł, zapomocą których mógłbym poznać, która liczba jest pierwszą, [6]a jakie są cechy liczb złożonych. Przy pomocy Dra Hossfelda w Eisenach, poznałem zasady funkcyi $\operatorname {\varphi } (m)$. Opracowując tę funkcyę w zakresie do $100\,000$, potem do $1,000\,000$, wykryłem okresy symetrycznych luk między liczbami względnie pierwszemi, po usunięciu każdej z porządku z liczb pierwszych ze wszystkiemi przez nią podzielnemi liczbami i prawidło ogólne, że

	$\operatorname {\varphi } (p_{1}.p_{2}.p_{3}\ldots p_{n},n)=(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1)$
oraz że	$\operatorname {\varphi } (g(p_{1}.p_{2}.p_{3}\ldots p_{n},n))=g(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1)$.

 Wykryta zaś symetrya luk w danym okresie, t. j. że luka 1-sza $=$ luce ostatniej, 2-ga $=$ przedostatniej , 3-cia z początku $=$ 3-ciej od końca i t. d. aż do połowy, czyli środka okresu, to jest do ${{p_{1}.p_{2}.p_{3}\ldots p_{n}} \over 2}$ wskazała możność obliczenia na poczekaniu i zakresu numeracyi
$0,1,2,3,\ldots m=p_{1}.p_{2}.p_{3}\ldots p_{n}+r$,
oraz $m=g.p_{1}.p_{2}.p_{3}\ldots p_{n}\pm {r}$,
gdyż $\operatorname {\varphi } (p_{1}.p_{2}.p_{3}\ldots p_{n}\pm {r},n)=(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1){\begin{cases}+\operatorname {\varphi } (r,n)\\-\operatorname {\varphi } (r-1,n),\end{cases}}$
oraz $\operatorname {\varphi } (g.(p_{1}.p_{2}.p_{3}\ldots p_{n}\pm {r},n))=(g(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1){\begin{cases}+\operatorname {\varphi } (r,n)\\-\operatorname {\varphi } (r-1,n).\end{cases}}$
 Zakomunikowałem te spostrzeżenia przez p. Webera Drowi Hossfeldowi, który przyznał im wielką wagę; ale potem zawiadomił mię, że już w roku 1872 Dr. Meissel te same prawa odkrył i ogłosił, tylko jeszcze nie wiedział o wewnętrznej symetryi okresów i dla tego wzór jego posiada tylko $+r,$ ale nie $-r.$ Wydrukował o tem za mojem przyzwoleniem króciutki artykulik o dopełnieniu przeze mnie wzoru Meissela. Artykulik wyszedł niefortunnie, bo przed wydrukowaniem nie przysłał mi go do przejrzenia. Wsadził doń bez potrzeby własnego pomysłu $p_{x},$ bez potrzeby i sensu. Kiedym go później przekonał, iż wszystkie trudności i pomyłki wyrażenie $-\operatorname {\varphi } (r-1,n)$ usuwa, przysłał mi do przejrzenia napisany do druku drugi artykulik o tem samem; ale jego objaśnienie $-\operatorname {\varphi } (r-1,n),-1$, nie trafiło do mojego przekonania; moje zaś objaśnienie on uważał za naciągnięte. W ten sposób artykulik jego nie wyszedł z druku.
 Dla większego jeszcze uproszczenia obliczeń funkcyi $\operatorname {\varphi } (m)$, ułożyłem analityczną tablicę okresu $p_{6},$ t. j. $\operatorname {\varphi } (2.3.5.7.11.13,6)=\operatorname {\varphi } (30030,6)$, która wielkie daje ułatwienie przy obliczaniu większych zakresów numeracyi.
[7] Potem napisałem teoryę luk, których Dr. Hossfeld nie podjął się sprawdzić.
 Nareszcie usunąwszy wszystkie liczby podzielne przez $2,$ $3$ i $5$, jako łatwe do poznania, zanalizowałem zakres numeracyi $0,$ $1,$ $2,$ $3,\ldots$ $150\,060$, czyli liczby wszystkie pierwsze w tym zakresie, oraz podzielne przez $7,$ $11,$ $13,$ $17,\ldots ,$ t. j. przez $p_{4},$ $p_{5},$ $p_{6},$ $p_{7},\ldots$ Spisałem na ogół liczb $40\,008$, oznaczając je właściwemi czynnikami n. p. $49=7^{2},$ $77=7.11;$ $91=7.13,$ $1001=7.11.13$ i t. d.
 Później z tego kajetu wypisałem osobno same tylko liczby pierwsze, to jest:

$p_{1}=2$, $p_{2}=3$, $p_{3}=5$, $p_{4}=7$ .... $p_{13852}=150\,053$.

 Gustaw Wertheim w dziele „Elemente der Zahlentheorie“ (Leipzig 1887) rozwija i przykładem objaśnia następujący wzór Meissel’a do obliczenia w danym zakresie numeracyi liczb pierwszych str. 24.

$\operatorname {\psi } (n)=\operatorname {\varphi } (n,m)+m(\mu +1)+{{\mu (\mu -1)} \over 2}-1-\sum _{s=1}^{s=\mu }\operatorname {\psi } {\biggl (}{n \over {p_{m+s}}}{\biggr )}$

$\operatorname {\psi } (n)$ oznacza tutaj, ile się zawiera liczb bezwzględnie pierwszych w zakresie numeracyi od $0,$ $1,$ $2,$ $3,\ldots$ $n.$
 $m$ oznacza, ile liczb pierwszych znajduje się w sześciennym pierwiastku zakresu $n$, czyli $m=\operatorname {\varphi } {\Bigl (}{\sqrt[{3}]{n}}{\Bigr )}$.
 $\mu$ oznacza, ile liczb pierwszych znajduje się w pierwiastku kwadratowym ${\sqrt {2}}$ , po odjęciu liczby tychże liczb, będących w pierwiastku sześciennym, czyli $\mu =\psi {\sqrt {n}}-m$.

$m+\mu =\psi {\Bigl (}{\sqrt {n}}{\Bigr )}$.

 Wzór ten, dobry przy obliczaniu niewielkich zakresów numeracyi, kiedy $n$ nie przewyższa setek, tysięcy; znośny jeszcze i przy obliczaniu dziesiątek tysięcy; w wielkich zaś zakresach numeracyi, wymaga wiele miejsca, czasu i pracy. Można się o tem przekonać, obliczając choćby tylko

$n=100\,000$; ${\sqrt[{3}]{n}}=46$; $\operatorname {\psi } (46)=14$; ${\sqrt {n}}=316$; $\operatorname {\psi } (316)=65$;

ponieważ zaś $m=\operatorname {\psi } ({\sqrt[{3}]{n}})=\operatorname {\psi } (46)=14,$ [8]przeto $\mu =\left[\psi \left({\sqrt {n}}\right)=\psi \left(316\right)=65\right]-14=51;$ $m+\mu =\psi \left({\sqrt {n}}\right)=65$
$\varphi (n,m)=\varphi (100\,000,14)=14204$. Tego trzeba dowieść rachunkiem następującym.
 Liczby te 14 są następujące: $p_{1}=2,$ $p_{2}=3,$ $p_{3}=5,$ $p_{4}=7,$ $p_{5}=11,$ $p_{6}=13,$ $p_{7}=17,$ $p_{8}=19,$ $p_{9}=23,$ $p_{10}=29,$ $p_{11}=31,$ $p_{12}=37,$ $p_{13}=41,$ $p_{14}=43;$

$\varphi (100\,000,14)=\varphi (100\,000,13)-\left[\varphi \left({\frac {100\,000}{43}},13\right)=\varphi \left(\,\,\,\,2325,13\right)\right]=14\,540-\,\,\,\,\,\,\,336=14\,204\,\,$

$\varphi (100\,000,13)=\varphi (100\,000,12)-\left[\varphi \left({\frac {100\,000}{41}},12\right)=\varphi \left(\,\,\,\,2439,12\right)\right]=14\,899-\,\,\,\,\,\,\,359=14\,540\,\,$

$\varphi (100\,000,12)=\varphi (100\,000,11)-\left[\varphi \left({\frac {100\,000}{37}},11\right)=\varphi \left(\,\,\,\,2707,11\right)\right]=15\,305-\,\,\,\,\,\,\,406=14\,899\,\,$

$\varphi (100\,000,11)=\varphi (100\,000,10)-\left[\varphi \left({\frac {100\,000}{31}},10\right)=\varphi \left(\,\,\,\,3225,10\right)\right]=15\,805-\,\,\,\,\,\,\,500=15\,305\,\,$

$\varphi (100\,000,10)=\varphi (100\,000,\,\,\,9)-\left[\varphi \left({\frac {100\,000}{29}},\,\,\,9\right)=\varphi \left(\,\,\,\,3448,\,\,\,9\right)\right]=16\,361-\,\,\,\,\,\,\,556=15\,805\,\,$

$\varphi (100\,000,\,\,\,9)=\varphi (100\,000,\,\,\,8)-\left[\varphi \left({\frac {100\,000}{23}},\,\,\,8\right)=\varphi \left(\,\,\,\,4347,\,\,\,8\right)\right]=17\,103-\,\,\,\,\,\,\,742=16\,361\,\,$

$\varphi (100\,000,\,\,\,8)=\varphi (100\,000,\,\,\,7)-\left[\varphi \left({\frac {100\,000}{19}},\,\,\,7\right)=\varphi \left(\,\,\,\,5263,\,\,\,7\right)\right]=18\,053-\,\,\,\,\,\,\,950=17\,103\,\,$

$\varphi (100\,000,\,\,\,7)=\varphi (100\,000,\,\,\,6)-\left[\varphi \left({\frac {100\,000}{17}},\,\,\,6\right)=\varphi \left(\,\,\,\,5882,\,\,\,6\right)\right]=19\,181-\,\,\,\,1128=18\,053\,\,$

$\varphi (100\,000,\,\,\,6)=\varphi (100\,000,\,\,\,5)-\left[\varphi \left({\frac {100\,000}{13}},\,\,\,5\right)=\varphi \left(\,\,\,\,7692,\,\,\,5\right)\right]=20\,779-\,\,\,\,1598=19\,181\,\,$

$\varphi (100\,000,\,\,\,5)=\varphi (100\,000,\,\,\,4)-\left[\varphi \left({\frac {100\,000}{11}},\,\,\,4\right)=\varphi \left(\,\,\,\,9090,\,\,\,4\right)\right]=22\,857-\,\,\,\,2078=20\,779\,\,$

[9]

$\varphi (100\,000,\,\,\,4)=\varphi (100\,000,\,\,\,3)-\left[\varphi \left({\frac {100\,000}{7}},\,\,\,3\right)=\varphi \left(14\,285,\,\,\,3\right)\right]=26\,666-\,\,\,\,3809=22\,857\,\,$

$\varphi (100\,000,\,\,\,3)=\varphi (100\,000,\,\,\,2)-\left[\varphi \left({\frac {100\,000}{5}},\,\,\,2\right)=\varphi \left(20\,000,\,\,\,2\right)\right]=33\,333-\,\,\,\,6667=26\,666\,\,$

$\varphi (100\,000,\,\,\,2)=\varphi (100\,000,\,\,\,1)-\left[\varphi \left({\frac {100\,000}{3}},\,\,\,1\right)=\varphi \left(33\,333,\,\,\,1\right)\right]=50\,000-16\,667=33\,333.$

[10] [11] [12] [13] [14]

$\varphi (20\,000,2)=\varphi (20\,000,1)-\varphi (6\,666,1)=10\,000-3\,333$	$=$	$6\,667$
$\varphi (14\,285,3)=\varphi (14\,285,2)-\varphi (2\,857,2)=4\,762-953=3\,809$ $\varphi (14\,285,2)=\varphi (14\,285,1)-\varphi (4\,761,1)=7\,143-2\,381=4\,762$ $\varphi (2857,2)=\varphi (2\,857,1)-\varphi (952,1)=1\,429-476$	$=$	$953$
$\varphi (9\,090,4)=\varphi (9\,090,3)-\varphi (1\,298,3)=2\,424-346=2\,078$ $\varphi (9\,090,3)=\varphi (9\,090,2)-\varphi (1\,818,2)=3\,030-606=2\,424$ $\varphi (9\,090,2)=\varphi (9\,090,1)-\varphi (3\,030,1)=4\,545-1\,515=3\,030$ $\varphi (1\,818,2)=\varphi (1\,818,1)-\varphi (606,1)=909-303$	$=$	$606$
$\varphi (1\,298,3)=\varphi (1\,298,2)-\varphi (259,2)=433-87=346$ $\varphi (1\,298,2)=\varphi (1\,298,1)-\varphi (432,1)=649-216=433$ $\varphi (259,2)=\varphi (259,1)-\varphi (86,1)=130-43$	$=$	$87$
$\varphi (7\,692,5)=\varphi (7\,692,4)-\varphi (699,4)=1\,758-160=1\,598$ $\varphi (7\,692,4)=\varphi (7\,692,3)-\varphi (1\,098,3)=2\,051-293=1\,758$ $\varphi (7\,692,3)=\varphi (7\,692,2)-\varphi (1\,538,2)=2\,564-513=2\,051$ $\varphi (7\,692,2)=\varphi (7\,692,1)-\varphi (2\,564,1)=3\,846-1\,282=2\,564$ $\varphi (1\,538,2)=\varphi (1\,538,1)-\varphi (512,1)=769-256$	$=$	$513$
$\varphi (1\,098,3)=\varphi (1\,098,2)-\varphi (219,2)=366-73=293$ $\varphi (1\,098,2)=\varphi (1\,098,1)-\varphi (366,1)=549-183=366$ $\varphi (219,2)=\varphi (219,1)-\varphi (73,1)=110-37$	$=$	$73$
$\varphi (699,4)=\varphi (699,3)-\varphi (99,3)=186-26=160$ $\varphi (699,3)=\varphi (699,2)-\varphi (139,2)=233-47=186$ $\varphi (699,2)=\varphi (699,1)-\varphi (233,1)=350-117=233$ $\varphi (139,2)=\varphi (139,1)-\varphi (46,1)=70-23$	$=$	$47$
$\varphi (99,3)=\varphi (99,2)-\varphi (19,2)=33-7=26$ $\varphi (99,2)=\varphi (99,1)-\varphi (33,1)=50-17=33$ $\varphi (19,2)=\varphi (19,1)-\varphi (6,1)=10-3$	$=$	$7$
$\varphi (5882,6)=\varphi (5882,5)-\varphi (452,5)=1222-94=1128$ $\varphi (5882,5)=\varphi (5882,4)-\varphi (534,4)=1345-123=1222$ $\varphi (5882,4)=\varphi (5882,3)-\varphi (840,3)=1569-224=1345$ $\varphi (5882,3)=\varphi (5882,2)-\varphi (1176,2)=1961-392=1569$ $\varphi (5882,2)=\varphi (5882,1)-\varphi (1960,1)=2941-980=1961$ $\varphi (1176,2)=\varphi (1176,1)-\varphi (391,1)=588-196$	$=$	$392$
$\varphi (840,3)=\varphi (840,2)-\varphi (168,2)=280-56=224$ $\varphi (840,2)=\varphi (840,1)-\varphi (280,1)=420-140=280$ $\varphi (168,2)=\varphi (168,1)-\varphi (56,1)=84-28$	$=$	$56$
$\varphi (534,4)=\varphi (534,3)-\varphi (76,3)=143-20=123$ $\varphi (534,3)=\varphi (534,2)-\varphi (106,2)=178-35=143$ $\varphi (534,2)=\varphi (534,1)-\varphi (178,1)=267-89=178$ $\varphi (106,2)=\varphi (106,1)-\varphi (35,1)=53-18$	$=$	$35$
$\varphi (76,3)=\varphi (76,2)-\varphi (15,2)=25-5=20$ $\varphi (76,2)=\varphi (76,1)-\varphi (25,1)=38-13=25$ $\varphi (15,2)=\varphi (15,1)-\varphi (5,1)=8-3$	$=$	$5$
$\varphi (452,5)=\varphi (452,4)-\varphi (41,4)=104-10=94$ $\varphi (452,4)=\varphi (452,3)-\varphi (64,3)=121-17=104$ $\varphi (452,3)=\varphi (452,2)-\varphi (90,2)=151-30=121$ $\varphi (452,2)=\varphi (452,1)-\varphi (150,1)=226-75=151$ $\varphi (90,2)=\varphi (90,1)-\varphi (90,1)=45-15$	$=$	$30$
$\varphi (64,3)=\varphi (64,2)-\varphi (12,2)=21-4=17$ $\varphi (64,2)=\varphi (64,1)-\varphi (12,1)=32-11=21$ $\varphi (12,2)=\varphi (12,1)-\varphi (4,1)=6-2$	$=$	$4$
$\varphi (41,4)=\varphi (41,3)-\varphi (5,3)=11-1=10$ $\varphi (41,3)=\varphi (41,2)-\varphi (8,2)=14-3=11$ $\varphi (41,2)=\varphi (41,1)-\varphi (13,1)=21-7=14$
$\varphi (5263,7)=\varphi (5263,6)-\varphi (309,6)=1009-59=950$ $\varphi (5263,6)=\varphi (5263,5)-\varphi (404,5)=1094-85=1009$ $\varphi (5263,5)=\varphi (5263,4)-\varphi (478,4)=1203-109=1094$ $\varphi (5263,4)=\varphi (5263,3)-\varphi (751,3)=1404-201=1203$ $\varphi (5263,3)=\varphi (5263,2)-\varphi (1052,2)=1755-351=1404$ $\varphi (5263,2)=\varphi (5263,1)-\varphi (1754,1)=2632-877=1755$ $\varphi (1052,2)=\varphi (1052,1)-\varphi (350,1)=526-175$	$=$	$351$
$\varphi (751,3)=\varphi (751,2)-\varphi (150,2)=251-\,\,\,50=201$ $\varphi (751,2)=\varphi (751,1)-\varphi (250,1)=376-125=251$ $\varphi (150,2)=\varphi (150,1)-\varphi (50,1)=75-25$	$=$	$50$
$\varphi (478,4)=\varphi (478,3)-\varphi (68,3)=127-18=109$ $\varphi (478,3)=\varphi (478,2)-\varphi (95,2)=159-32=127$ $\varphi (478,2)=\varphi (478,1)-\varphi (159,1)=239-80=159$ $\varphi (95,2)=\varphi (95,1)-\varphi (31,1)=48-16$	$=$	$32$
$\varphi (68,3)=\varphi (68,2)-\varphi (13,2)=23-5=18$ $\varphi (68,2)=\varphi (68,1)-\varphi (22,1)=34-11=23$ $\varphi (13,2)=\varphi (13,1)-\varphi (4,1)=7-2$	$=$	$5$
$\varphi (404,5)=\varphi (404,4)-\varphi (36,4)=93-8=85$ $\varphi (404,4)=\varphi (404,3)-\varphi (57,3)=108-15=93$ $\varphi (404,3)=\varphi (404,2)-\varphi (80,2)=135-27=108$ $\varphi (404,2)=\varphi (404,1)-\varphi (134,1)=202-67=135$ $\varphi (80,2)=\varphi (80,1)-\varphi (26,1)=40-13$	$=$	$27$
$\varphi (57,3)=\varphi (57,2)-\varphi (11,2)=19-4=15$ $\varphi (57,2)=\varphi (57,1)-\varphi (19,1)=29-10=19$ $\varphi (11,2)=\varphi (11,1)-\varphi (3,1)=6-2$	$=$	$4$
$\varphi (36,4)=\varphi (36,3)-\varphi (5,3)=9-1=8$ $\varphi (36,3)=\varphi (36,2)-\varphi (7,2)=12-3=9$ $\varphi (36,2)=\varphi (36,1)-\varphi (12,1)=18-6=12$ $\varphi (7,2)=\varphi (7,1)-\varphi (2,1)=4-1$	$=$	$3$
$\varphi (5,3)=\varphi (5,2)-\varphi (1,2)=2-1=1$ $\varphi (5,2)=\varphi (5,1)-\varphi (1,1)=3-1=2$
$\varphi (309,6)=\varphi (309,5)-\varphi (23,5)=64-5=59$ $\varphi (309,5)=\varphi (309,4)-\varphi (28,4)=70-6=64$ $\varphi (309,4)=\varphi (309,3)-\varphi (44,3)=82-12=70$ $\varphi (309,3)=\varphi (309,2)-\varphi (61,2)=103-21=82$ $\varphi (309,2)=\varphi (309,1)-\varphi (103,1)=155-52=103$ $\varphi (61,2)=\varphi (61,1)-\varphi (20,1)=31-10$	$=$	$21$
$\varphi (44,3)=\varphi (44,2)-\varphi (8,2)=15-3=12$ $\varphi (44,2)=\varphi (44,1)-\varphi (14,1)=22-7=15$ $\varphi (8,2)=\varphi (8,1)-\varphi (2,1)=4-1$	$=$	$3$
$\varphi (25,4)=\varphi (28,3)-\varphi (4,3)=7-1=6$ $\varphi (28,3)=\varphi (28,2)-\varphi (5,2)=9-2=7$ $\varphi (28,2)=\varphi (28,1)-\varphi (9,1)=14-5=9$ $\varphi (5,2)=\varphi (5,1)-\varphi (1,1)=3-1$	$=$	$2$
$\varphi (23,5)=\varphi (23,4)-\varphi (2,4)=\,\,\,6-1=5$ $\varphi (23,4)=\varphi (23,3)-\varphi (3,3)=\,\,\,7-1=6$ $\varphi (23,3)=\varphi (23,2)-\varphi (4,2)=\,\,\,8-1=7$ $\varphi (23,2)=\varphi (23,1)-\varphi (7,1)=12-4=8$	${\displaystyle }$	${\displaystyle }$
$\varphi (4347,8)=\varphi (4347,7)-\varphi (\,\,\,228,7)=\,\,\,785-\,\,\,43=\,\,\,742$ $\varphi (4347,7)=\varphi (4347,6)-\varphi (\,\,\,255,6)=\,\,\,834-\,\,\,49=\,\,\,785$ $\varphi (4347,6)=\varphi (4347,5)-\varphi (\,\,\,334,5)=\,\,\,903-\,\,\,69=\,\,\,834$ $\varphi (4347,5)=\varphi (4347,4)-\varphi (\,\,\,395,4)=\,\,\,993-\,\,\,90=\,\,\,903$ $\varphi (4347,4)=\varphi (4347,3)-\varphi (\,\,\,621,3)=1159-166=\,\,\,993$ $\varphi (4347,3)=\varphi (4347,2)-\varphi (\,\,\,869,2)=1149-290=1159$ $\varphi (4347,2)=\varphi (4347,1)-\varphi (1449,1)=2174-725=1449$ $\varphi (869,2)=\varphi (869,1)-\varphi (289,1)=435-\,\,\,145$	$=$	$290$
$\varphi (621,3)=\varphi (621,2)-\varphi (124,2)=207-\,\,\,41$	$=$	$164$
$\varphi (621,2)=\varphi (621,1)-\varphi (207,1)=311-104$	$=$	$207$
$\varphi (124,2)=\varphi (124,1)-\varphi (41,1)=\,\,\,62-\,\,\,21$	$=$	$41$
$\varphi (395,4)=\varphi (395,3)-\varphi (56,3)=105-15=\,\,\,90$ $\varphi (395,3)=\varphi (395,2)-\varphi (79,2)=132-27=105$ $\varphi (395,2)=\varphi (395,1)-\varphi (131,1)=198-66=132$ $\varphi (79,2)=\varphi (79,1)-\varphi (26,1)=40-13$	$=$	$27$
$\varphi (56,3)=\varphi (56,2)-\varphi (11,2)=19-4=15$ $\varphi (56,2)=\varphi (56,1)-\varphi (18,1)=28-9=19$ $\varphi (11,2)=\varphi (11,1)-\varphi (3,1)=6-\,\,\,2$	$=$	$4$
$\varphi (334,5)=\varphi (334,4)-\varphi (\,\,\,30,4)=\,\,\,76-\,\,\,7=\,\,\,69$ $\varphi (334,4)=\varphi (334,3)-\varphi (\,\,\,47,3)=\,\,\,89-13=\,\,\,76$ $\varphi (334,3)=\varphi (334,2)-\varphi (\,\,\,66,2)=111-22=\,\,\,89$ $\varphi (334,2)=\varphi (334,1)-\varphi (111,1)=167-56=111$ $\varphi (66,2)=\varphi (66,1)-\varphi (22,1)=33-\,\,\,11$	$=$	$22$
$\varphi (47,3)=\varphi (47,2)-\varphi (\,\,\,9,2)=\,\,\,\,\,\,\,\,\,\,\,16-\,\,\,\,\,\,\,\,\,\,\,3$	$=$	$13$
$\varphi (47,2)=\varphi (47,1)-\varphi (15,1)=\,\,\,\,\,\,\,\,\,\,\,24-\,\,\,\,\,\,\,\,\,\,\,8$	$=$	$16$
$\varphi (\,\,\,9,2)=\varphi (\,\,\,9,1)-\varphi (3,1)=5-2$	$=$	$\,\,\,3$
$\varphi (30,4)=\varphi (30,3)-\varphi (\,\,\,4,3)=\,\,\,8-1=\,\,\,7$ $\varphi (30,3)=\varphi (30,2)-\varphi (\,\,\,6,2)=10-2=\,\,\,8$ $\varphi (30,2)=\varphi (30,1)-\varphi (10,1)=15-5=10$	${\displaystyle }$	${\displaystyle }$
$\varphi (255,6)=\varphi (255,5)-\varphi (19,5)=\,\,\,\,\,\,53\,\,\,\,\,\,\,-\,\,\,4=49$  $\varphi (255,5)=\varphi (255,4)-\varphi (23,4)=\,\,\,\,\,\,59\,\,\,\,\,\,\,-\,\,\,6=53$  $\varphi (255,4)=\varphi (255,3)-\varphi (36,3)=\,\,\,\,\,\,68\,\,\,\,\,\,\,-\,\,\,9=59$  $\varphi (255,3)=\varphi (255,2)-\varphi (51,2)=\,\,\,\,\,\,85\,\,\,\,\,\,\,-17=68$  $\varphi (255,2)=\varphi (255,1)-\varphi (85,1)=\,\,\,128\,\,\,\,\,\,\,-43=85$  $\varphi (\,\,\,51,2)=\varphi (51,1)-\varphi (17,1)=26-\,\,\,9$	$=$	$17$
$\varphi (36,3)=\varphi (36,2)-\varphi (\,\,\,7,2)=\,\,\,\,\,\,\,\,\,\,\,12-3=\,\,\,9$ $\varphi (36,2)=\varphi (36,1)-\varphi (12,1)=\,\,\,\,\,\,\,\,\,\,\,18-6=12$ $\varphi (\,\,\,7,2)=\varphi (\,\,\,7,1)-\varphi (2,1)=4-\,\,\,1$	$=$	$3$
$\varphi (23,4)=\varphi (23,3)-\varphi (3,3)=\,\,\,7-1=6$ $\varphi (23,3)=\varphi (23,2)-\varphi (4,2)=\,\,\,8-1=7$ $\varphi (23,2)=\varphi (23,1)-\varphi (7,1)=12-4=8$
$\varphi (19,5)=\varphi (19,4)-\varphi (1,4)=\,\,\,5-1=4$ $\varphi (19,4)=\varphi (19,3)-\varphi (2,3)=\,\,\,6-1=5$ $\varphi (19,3)=\varphi (19,2)-\varphi (3,2)=\,\,\,7-1=6$ $\varphi (19,2)=\varphi (19,1)-\varphi (6,1)=10-3=7$
$\varphi (228,7)=\varphi (228,6)-\varphi (13,6)=\,\,\,\,\,\,44\,\,\,\,\,\,\,\,-\,\,\,1=43$  $\varphi (228,6)=\varphi (228,5)-\varphi (17,5)=\,\,\,\,\,\,47\,\,\,\,\,\,\,\,-\,\,\,3=44$  $\varphi (228,5)=\varphi (228,4)-\varphi (20,4)=\,\,\,\,\,\,52\,\,\,\,\,\,\,\,-\,\,\,5=47$  $\varphi (228,4)=\varphi (228,3)-\varphi (32,3)=\,\,\,\,\,\,61\,\,\,\,\,\,\,\,-\,\,\,9=52$  $\varphi (228,3)=\varphi (228,2)-\varphi (45,2)=\,\,\,\,\,\,76\,\,\,\,\,\,\,\,-15=61$  $\varphi (228,2)=\varphi (228,1)-\varphi (76,1)=\,\,\,114\,\,\,\,\,\,\,\,-38=76$  $\varphi (\,\,\,45,2)=\varphi (45,1)-\varphi (15,1)=23-\,\,\,8$	$=$	$15$
$\varphi (32,3)=\varphi (32,2)-\varphi (\,\,\,6,2)=11-2$	$=$	$9$
$\varphi (32,2)=\varphi (32,1)-\varphi (10,1)=16-5$	$=$	$11$
$\varphi (3448,9)=\varphi (3448,8)-\varphi (\,\,\,149,8)=\,\,\,\,\,\,584\,\,\,\,\,\,\,\,-\,\,\,28=\,\,\,556$ $\varphi (3448,8)=\varphi (3448,7)-\varphi (\,\,\,181,7)=\,\,\,\,\,\,620\,\,\,\,\,\,\,\,-\,\,\,36=\,\,\,584$ $\varphi (3448,7)=\varphi (3448,6)-\varphi (\,\,\,202,6)=\,\,\,\,\,\,661\,\,\,\,\,\,\,\,-\,\,\,41=\,\,\,620$ $\varphi (3448,6)=\varphi (3448,5)-\varphi (\,\,\,265,5)=\,\,\,\,\,\,716\,\,\,\,\,\,\,\,-\,\,\,55=\,\,\,661$ $\varphi (3448,5)=\varphi (3448,4)-\varphi (\,\,\,313,4)=\,\,\,\,\,\,788\,\,\,\,\,\,\,\,-\,\,\,72=\,\,\,716$ $\varphi (3448,4)=\varphi (3448,3)-\varphi (\,\,\,492,3)=\,\,\,\,\,\,919\,\,\,\,\,\,\,\,-131=\,\,\,788$ $\varphi (3448,3)=\varphi (3448,2)-\varphi (\,\,\,689,2)=\,\,\,1149\,\,\,\,\,\,\,\,-230=\,\,\,919$ $\varphi (3448,2)=\varphi (3448,1)-\varphi (1149,1)=\,\,\,1724\,\,\,\,\,\,\,\,-575=1149$ $\varphi (\,\,\,689,2)=\varphi (\,\,\,689,1)-\varphi (229,1)=345-\,\,\,115$	$=$	$230$
$\varphi (492,3)=\varphi (492,2)-\varphi (\,\,\,98,2)=164-33$	$=$	$131$
$\varphi (492,2)=\varphi (491,1)-\varphi (164,1)=246-82$	$=$	$164$
$\varphi (98,2)=\varphi (98,1)-\varphi (32,1)=49-16$	$=$	$33$

$\varphi (313,4)=\varphi (313,3)-\varphi (\,\,\,44,3)=\,\,\,\,\,\,84\,\,\,\,\,\,\,-12=\,\,\,72$  $\varphi (313,3)=\varphi (313,2)-\varphi (\,\,\,62,2)=\,\,\,105\,\,\,\,\,\,\,-21=\,\,\,84$  $\varphi (313,2)=\varphi (313,1)-\varphi (104,1)=\,\,\,157\,\,\,\,\,\,\,-52=105$  $\varphi (\,\,\,62,2)=\varphi (62,1)-\varphi (20,1)=31-\,\,\,10$	$=$	$21$
$\varphi (\,\,\,44,3)=\varphi (44,2)-\varphi (\,\,\,8,2)=15-\,\,\,\,\,\,3$  $\varphi (\,\,\,44,2)=\varphi (44,1)-\varphi (12,1)=22-\,\,\,\,\,\,7$	$=$ $=$	$12$ $15$
$\varphi (265,5)=\varphi (265,4)-\varphi (\,\,\,24,4)=\,\,\,\,\,\,61\,\,\,\,\,\,\,-\,\,\,6=\,\,\,55$  $\varphi (265,4)=\varphi (265,3)-\varphi (\,\,\,37,3)=\,\,\,\,\,\,71\,\,\,\,\,\,\,-10=\,\,\,61$  $\varphi (265,3)=\varphi (265,2)-\varphi (\,\,\,53,2)=\,\,\,\,\,\,89\,\,\,\,\,\,\,-18=\,\,\,71$  $\varphi (265,2)=\varphi (265,1)-\varphi (\,\,\,88,1)=\,\,\,133\,\,\,\,\,\,\,-44=\,\,\,89$  $\varphi (\,\,\,53,2)=\varphi (53,1)-\varphi (17,1)=27-\,\,\,\,\,\,9$	$=$	$18$
$\varphi (\,\,\,37,3)=\varphi (37,2)-\varphi (\,\,\,7,2)=13-\,\,\,\,\,\,3$  $\varphi (\,\,\,37,2)=\varphi (37,1)-\varphi (14,1)=19-\,\,\,\,\,\,6$	$=$ $=$	$10$ $13$
$\varphi (202,6)=\varphi (202,5)-\varphi (\,\,\,15,5)=\,\,\,\,\,\,43\,\,\,\,\,\,\,-\,\,\,2=\,\,\,41$  $\varphi (202,5)=\varphi (202,4)-\varphi (\,\,\,18,4)=\,\,\,\,\,\,47\,\,\,\,\,\,\,-\,\,\,4=\,\,\,43$  $\varphi (202,4)=\varphi (202,3)-\varphi (\,\,\,28,3)=\,\,\,\,\,\,54\,\,\,\,\,\,\,-\,\,\,7=\,\,\,47$  $\varphi (202,3)=\varphi (202,2)-\varphi (\,\,\,40,2)=\,\,\,\,\,\,67\,\,\,\,\,\,\,-13=\,\,\,54$  $\varphi (202,2)=\varphi (202,1)-\varphi (\,\,\,67,1)=\,\,\,101\,\,\,\,\,\,\,-34=\,\,\,67$
$\varphi (181,7)=\varphi (181,6)-\varphi (\,\,\,10,6)=\,\,\,\,\,\,37\,\,\,\,\,\,\,-\,\,\,1=\,\,\,36$  $\varphi (181,6)=\varphi (181,5)-\varphi (\,\,\,13,5)=\,\,\,\,\,\,39\,\,\,\,\,\,\,-\,\,\,2=\,\,\,37$  $\varphi (181,5)=\varphi (181,4)-\varphi (\,\,\,16,4)=\,\,\,\,\,\,42\,\,\,\,\,\,\,-\,\,\,3=\,\,\,39$  $\varphi (181,4)=\varphi (181,3)-\varphi (\,\,\,25,3)=\,\,\,\,\,\,49\,\,\,\,\,\,\,-\,\,\,7=\,\,\,42$  $\varphi (181,3)=\varphi (181,2)-\varphi (\,\,\,36,2)=\,\,\,\,\,\,61\,\,\,\,\,\,\,-12=\,\,\,49$  $\varphi (181,2)=\varphi (181,1)-\varphi (\,\,\,50,1)=\,\,\,\,\,\,91\,\,\,\,\,\,\,-30=\,\,\,61$  $\varphi (\,\,\,36,2)=\varphi (36,1)-\varphi (12,1)=18-\,\,\,\,\,\,6$	$=$	$12$
$\varphi (\,\,\,25,3)=\varphi (25,2)-\varphi (\,\,\,5,2)=\,\,\,9-\,\,\,\,\,\,2$  $\varphi (\,\,\,25,2)=\varphi (25,1)-\varphi (\,\,\,8,1)=13-\,\,\,\,\,\,4$	$=$ $=$	$8$ $9$
$\varphi (149,8)=\varphi (149,7)-\varphi (\,\,\,\,\,\,7,7)=\,\,\,\,\,\,29\,\,\,\,\,\,\,-\,\,\,1=\,\,\,28$  $\varphi (149,7)=\varphi (149,6)-\varphi (\,\,\,\,\,\,8,6)=\,\,\,\,\,\,30\,\,\,\,\,\,\,-\,\,\,1=\,\,\,29$  $\varphi (149,6)=\varphi (149,5)-\varphi (\,\,\,11,5)=\,\,\,\,\,\,31\,\,\,\,\,\,\,-\,\,\,1=\,\,\,30$  $\varphi (149,5)=\varphi (149,4)-\varphi (\,\,\,13,4)=\,\,\,\,\,\,34\,\,\,\,\,\,\,-\,\,\,3=\,\,\,31$  $\varphi (149,4)=\varphi (149,3)-\varphi (\,\,\,21,3)=\,\,\,\,\,\,40\,\,\,\,\,\,\,-\,\,\,6=\,\,\,34$  $\varphi (149,3)=\varphi (149,2)-\varphi (\,\,\,29,2)=\,\,\,\,\,\,50\,\,\,\,\,\,\,-10=\,\,\,40$  $\varphi (149,2)=\varphi (149,1)-\varphi (\,\,\,49,1)=\,\,\,\,\,\,75\,\,\,\,\,\,\,-25=\,\,\,50$  $\varphi (\,\,\,29,2)=\varphi (29,1)-\varphi (\,\,\,9,1)=15-\,\,\,\,\,\,5$	$=$	$10$
$\varphi (3225,10)=\varphi (3225,9)-\varphi (\,\,\,111,9)=\,\,\,\,\,\,521\,\,\,\,\,\,\,-21=\,\,\,500$  $\varphi (3225,\,\,\,9)=\varphi (3225,8)-\varphi (\,\,\,140,8)=\,\,\,\,\,\,548\,\,\,\,\,\,\,-27=\,\,\,521$  $\varphi (3225,\,\,\,8)=\varphi (3225,7)-\varphi (\,\,\,169,7)=\,\,\,\,\,\,581\,\,\,\,\,\,\,-33=\,\,\,548$  $\varphi (3225,\,\,\,7)=\varphi (3225,6)-\varphi (\,\,\,189,6)=\,\,\,\,\,\,618\,\,\,\,\,\,\,-37=\,\,\,581$  $\varphi (3225,\,\,\,6)=\varphi (3225,5)-\varphi (\,\,\,248,5)=\,\,\,\,\,\,670\,\,\,\,\,\,\,-52=\,\,\,618$  $\varphi (3225,\,\,\,5)=\varphi (3225,4)-\varphi (\,\,\,293,4)=\,\,\,\,\,\,738\,\,\,\,\,\,\,-68=\,\,\,670$  $\varphi (3225,\,\,\,4)=\varphi (3225,3)-\varphi (\,\,\,460,3)=\,\,\,\,\,\,860\,\,\,\,-122=\,\,\,738$  $\varphi (3225,\,\,\,3)=\varphi (3225,2)-\varphi (\,\,\,645,2)=\,\,\,1075\,\,\,\,-215=\,\,\,860$  $\varphi (3225,\,\,\,2)=\varphi (3225,1)-\varphi (1075,1)=\,\,\,1613\,\,\,\,-538=1075$  $\varphi (\,\,\,645,2)=\varphi (\,\,\,645,1)-\varphi (215,1)=323-\,\,\,108=215$
$\varphi (\,\,\,460,3)=\varphi (\,\,\,460,2)-\varphi (\,\,\,92,2)=153-\,\,\,\,\,\,31=122$  $\varphi (\,\,\,460,2)=\varphi (\,\,\,460,1)-\varphi (153,1)=230-\,\,\,\,\,\,77=153$  $\varphi (\,\,\,\,\,92,2)=\varphi (\,\,\,92,1)-\varphi (30,1)=46-15$	$=$	$\,\,\,31$
$\varphi (\,\,\,293,4)=\varphi (\,\,\,293,4)-\varphi (\,\,\,41,3)=\,\,\,79-\,\,\,\,\,\,11=\,\,\,68$  $\varphi (\,\,\,293,3)=\varphi (\,\,\,293,2)-\varphi (\,\,\,58,2)=\,\,\,98-\,\,\,\,\,\,19=\,\,\,79$  $\varphi (\,\,\,293,2)=\varphi (\,\,\,293,1)-\varphi (\,\,\,97,1)=147-\,\,\,\,\,\,49=\,\,\,98$  $\varphi (\,\,\,\,\,58,2)=\varphi (\,\,\,58,1)-\varphi (19,1)=19-10$	$=$	$\,\,\,19$
$\varphi (\,\,\,41,3)\,\,\,=\,\,\,\varphi (41,2)-\varphi (\,\,\,8,2)=41-\,\,\,3$  $\varphi (\,\,\,41,2)\,\,\,=\,\,\,\varphi (41,1)-\varphi (13,1)=21-\,\,\,7$  $\varphi (\,\,\,\,\,\,8,2)\,\,\,=\,\,\,\varphi (\,\,\,8,1)-\varphi (\,\,\,2,1)=\,\,\,4-\,\,\,1$	$=$ $=$ $=$	$\,\,\,11$ $\,\,\,14$ $\,\,\,\,\,\,3$
$\varphi (\,\,\,248,5)=\varphi (\,\,\,248,4)-\varphi (\,\,\,22,4)=\,\,\,57-\,\,\,\,\,\,5=52$  $\varphi (\,\,\,248,4)=\varphi (\,\,\,248,3)-\varphi (\,\,\,35,3)=\,\,\,66-\,\,\,\,\,\,9=57$  $\varphi (\,\,\,248,3)=\varphi (\,\,\,248,2)-\varphi (\,\,\,49,2)=\,\,\,83-\,\,\,17=66$  $\varphi (\,\,\,248,2)=\varphi (\,\,\,248,1)-\varphi (\,\,\,82,1)=124-\,\,\,41=83$  $\varphi (\,\,\,49,2)\,\,\,=\,\,\,\varphi (49,1)-\varphi (16,1)=25-\,\,\,8$	$=$	$\,\,\,17$
$\varphi (\,\,\,35,3)\,\,\,=\,\,\,\varphi (35,2)-\varphi (\,\,\,7,2)=12-\,\,\,3$  $\varphi (\,\,\,35,2)\,\,\,=\,\,\,\varphi (35,1)-\varphi (11,1)=18-\,\,\,6$  $\varphi (\,\,\,\,\,\,7,2)\,\,\,=\,\,\,\varphi (\,\,\,7,1)-\varphi (\,\,\,2,1)=\,\,\,4-\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,9$ $\,\,\,12$ $\,\,\,\,\,\,3$
$\varphi (\,\,\,189,6)=\varphi (\,\,\,189,5)-\varphi (\,\,\,14,5)=\,\,\,39-\,\,\,\,\,\,2=37$  $\varphi (\,\,\,189,5)=\varphi (\,\,\,189,4)-\varphi (\,\,\,17,4)=\,\,\,43-\,\,\,\,\,\,4=39$  $\varphi (\,\,\,189,4)=\varphi (\,\,\,189,3)-\varphi (\,\,\,27,3)=\,\,\,50-\,\,\,\,\,\,7=43$  $\varphi (\,\,\,189,3)=\varphi (\,\,\,189,2)-\varphi (\,\,\,37,2)=\,\,\,63-\,\,\,13=50$  $\varphi (\,\,\,189,2)=\varphi (\,\,\,189,1)-\varphi (\,\,\,63,1)=\,\,\,95-\,\,\,32=63$  $\varphi (\,\,\,37,2)\,\,\,=\,\,\,\varphi (37,1)-\varphi (12,1)=19-\,\,\,6$	$=$	$\,\,\,13$
$\varphi (\,\,\,27,3)=\varphi (\,\,\,27,2)-\varphi (\,\,\,\,\,\,5,2)=\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,2=\,\,\,\,\,\,7$  $\varphi (\,\,\,27,2)=\varphi (\,\,\,27,1)-\varphi (\,\,\,\,\,\,9,1)=\,\,\,14\,\,\,\,\,\,\,-\,\,\,5=\,\,\,\,\,\,9$  $\varphi (\,\,\,5,2)=\varphi (\,\,\,5,1)-\varphi (2,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$	$=$	$\,\,\,2$
$\varphi (\,\,\,17,4)=\varphi (\,\,\,17,3)-\varphi (\,\,\,\,\,\,1,3)=\,\,\,\,\,\,5\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,4$  $\varphi (\,\,\,17,3)=\varphi (\,\,\,17,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,6\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,5$  $\varphi (\,\,\,17,2)=\varphi (\,\,\,17,1)-\varphi (\,\,\,\,\,\,5,1)=\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,3=\,\,\,\,\,\,6$
$\varphi (169,7)=\varphi (169,6)-\varphi (\,\,\,\,\,\,9,6)=\,\,\,34\,\,\,\,\,\,\,-\,\,\,1=\,\,\,33$  $\varphi (169,6)=\varphi (169,5)-\varphi (\,\,\,13,5)=\,\,\,36\,\,\,\,\,\,\,-\,\,\,2=\,\,\,34$  $\varphi (169,5)=\varphi (169,4)-\varphi (\,\,\,15,4)=\,\,\,39\,\,\,\,\,\,\,-\,\,\,3=\,\,\,36$  $\varphi (169,4)=\varphi (169,3)-\varphi (\,\,\,24,3)=\,\,\,46\,\,\,\,\,\,\,-\,\,\,7=\,\,\,39$  $\varphi (169,3)=\varphi (169,2)-\varphi (\,\,\,33,2)=\,\,\,57\,\,\,\,\,\,\,-11=\,\,\,46$  $\varphi (169,2)=\varphi (169,1)-\varphi (\,\,\,56,1)=\,\,\,85\,\,\,\,\,\,\,-28=\,\,\,57$  $\varphi (33,2)=\varphi (33,1)-\varphi (11,1)=\,\,\,17-\,\,\,\,\,\,6$	$=$	$11$
$\varphi (\,\,\,24,3)=\varphi (\,\,\,24,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,8\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,7$  $\varphi (\,\,\,24,2)=\varphi (\,\,\,24,1)-\varphi (\,\,\,\,\,\,8,1)=\,\,\,12\,\,\,\,\,\,\,-\,\,\,4=\,\,\,\,\,\,8$
$\varphi (140,8)=\varphi (140,7)-\varphi (\,\,\,\,\,\,7,7)=\,\,\,28\,\,\,\,\,\,\,-\,\,\,1=\,\,\,27$  $\varphi (140,7)=\varphi (140,6)-\varphi (\,\,\,\,\,\,8,6)=\,\,\,29\,\,\,\,\,\,\,-\,\,\,1=\,\,\,28$  $\varphi (140,6)=\varphi (140,5)-\varphi (\,\,\,10,5)=\,\,\,30\,\,\,\,\,\,\,-\,\,\,1=\,\,\,29$  $\varphi (140,5)=\varphi (140,4)-\varphi (\,\,\,12,4)=\,\,\,32\,\,\,\,\,\,\,-\,\,\,2=\,\,\,30$  $\varphi (140,4)=\varphi (140,3)-\varphi (\,\,\,20,3)=\,\,\,38\,\,\,\,\,\,\,-\,\,\,6=\,\,\,32$  $\varphi (140,3)=\varphi (140,2)-\varphi (\,\,\,28,2)=\,\,\,47\,\,\,\,\,\,\,-\,\,\,9=\,\,\,38$  $\varphi (140,2)=\varphi (140,1)-\varphi (\,\,\,46,1)=\,\,\,70\,\,\,\,\,\,\,-23=\,\,\,47$  $\varphi (\,\,\,28,2)=\varphi (\,\,\,28,1)-\varphi (\,\,\,\,\,\,9,1)=\,\,\,14\,\,\,\,\,\,\,-\,\,\,5=\,\,\,\,\,\,9$
$\varphi (\,\,\,20,3)=\varphi (\,\,\,20,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,7\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,6$  $\varphi (\,\,\,20,2)=\varphi (\,\,\,20,1)-\varphi (\,\,\,\,\,\,6,1)=\,\,\,10\,\,\,\,\,\,\,-\,\,\,3=\,\,\,\,\,\,7$  $\varphi (\,\,\,4,2)=\varphi (\,\,\,4,1)-\varphi (\,\,\,1,1)=\,\,\,\,\,\,2-\,\,\,\,\,\,1$	$=$	$\,\,\,1$
$\varphi (\,\,\,12,4)=\varphi (\,\,\,12,3)-\varphi (\,\,\,\,\,\,1,3)=\,\,\,\,\,\,3\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,2$  $\varphi (\,\,\,12,3)=\varphi (\,\,\,12,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,4\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,3$  $\varphi (\,\,\,12,2)=\varphi (\,\,\,12,1)-\varphi (\,\,\,\,\,\,4,1)=\,\,\,\,\,\,6\,\,\,\,\,\,\,-\,\,\,2=\,\,\,\,\,\,4$
$\varphi (111,9)=\varphi (111,8)-\varphi (\,\,\,\,\,\,4,8)=\,\,\,22\,\,\,\,\,\,\,-\,\,\,1=\,\,\,21$  $\varphi (111,8)=\varphi (111,7)-\varphi (\,\,\,\,\,\,5,7)=\,\,\,23\,\,\,\,\,\,\,-\,\,\,1=\,\,\,22$  $\varphi (111,7)=\varphi (111,6)-\varphi (\,\,\,\,\,\,6,6)=\,\,\,24\,\,\,\,\,\,\,-\,\,\,1=\,\,\,23$  $\varphi (111,6)=\varphi (111,5)-\varphi (\,\,\,\,\,\,8,5)=\,\,\,25\,\,\,\,\,\,\,-\,\,\,1=\,\,\,24$  $\varphi (111,5)=\varphi (111,4)-\varphi (\,\,\,10,4)=\,\,\,26\,\,\,\,\,\,\,-\,\,\,1=\,\,\,25$  $\varphi (111,4)=\varphi (111,3)-\varphi (\,\,\,15,3)=\,\,\,30\,\,\,\,\,\,\,-\,\,\,4=\,\,\,26$  $\varphi (111,3)=\varphi (111,2)-\varphi (\,\,\,22,2)=\,\,\,37\,\,\,\,\,\,\,-\,\,\,7=\,\,\,30$  $\varphi (111,2)=\varphi (111,1)-\varphi (\,\,\,37,1)=\,\,\,56\,\,\,\,\,\,\,-19=\,\,\,37$  $\varphi (22,2)=\varphi (22,1)-\varphi (\,\,\,7,1)=\,\,\,11-\,\,\,\,\,\,4$	$=$	$\,\,\,7$
$\varphi (2702,11)=\varphi (2702,10)-\varphi (\,\,\,87,10)=\,\,\,\,\,\,420\,\,\,\,\,\,\,-\,\,\,14=\,\,\,406$  $\varphi (2702,10)=\varphi (2702,\,\,\,9)-\varphi (\,\,\,93,\,\,\,9)=\,\,\,\,\,\,436\,\,\,\,\,\,\,-\,\,\,16=\,\,\,420$  $\varphi (2702,\,\,\,9)=\varphi (2702,\,\,\,8)-\varphi (117,\,\,\,8)=\,\,\,\,\,\,459\,\,\,\,\,\,\,-\,\,\,23=\,\,\,436$  $\varphi (2702,\,\,\,8)=\varphi (2702,\,\,\,7)-\varphi (142,\,\,\,7)=\,\,\,\,\,\,487\,\,\,\,\,\,\,-\,\,\,28=\,\,\,459$  $\varphi (2702,\,\,\,7)=\varphi (2702,\,\,\,6)-\varphi (158,\,\,\,6)=\,\,\,\,\,\,519\,\,\,\,\,\,\,-\,\,\,32=\,\,\,487$  $\varphi (2702,\,\,\,6)=\varphi (2702,\,\,\,5)-\varphi (207,\,\,\,5)=\,\,\,\,\,\,562\,\,\,\,\,\,\,-\,\,\,43=\,\,\,519$  $\varphi (2702,\,\,\,5)=\varphi (2702,\,\,\,4)-\varphi (245,\,\,\,4)=\,\,\,\,\,\,618\,\,\,\,\,\,\,-\,\,\,56=\,\,\,562$  $\varphi (2702,\,\,\,4)=\varphi (2702,\,\,\,3)-\varphi (386,\,\,\,3)=\,\,\,\,\,\,721\,\,\,\,\,\,\,-103=\,\,\,618$  $\varphi (2702,\,\,\,3)=\varphi (2702,\,\,\,2)-\varphi (540,\,\,\,2)=\,\,\,\,\,\,901\,\,\,\,\,\,\,-180=\,\,\,721$  $\varphi (2702,\,\,\,2)=\varphi (2702,\,\,\,1)-\varphi (900,\,\,\,1)=\,\,\,1351\,\,\,\,\,\,\,-450=\,\,\,901$  $\varphi (\,\,\,540,\,\,\,2)=\varphi (540,\,\,\,1)-\varphi (180,1)=270-\,\,\,\,\,\,90$	$=$	$180$
$\varphi (386,\,\,\,3)=\varphi (386,\,\,\,2)-\varphi (\,\,\,77,2)=129-\,\,\,\,\,\,26$  $\varphi (386,\,\,\,2)=\varphi (386,\,\,\,1)-\varphi (128,1)=193-\,\,\,\,\,\,64$  $\varphi (\,\,\,77,\,\,\,2)=\varphi (\,\,\,77,\,\,\,1)-\varphi (\,\,\,25,1)=\,\,\,39-\,\,\,\,\,\,13$	$=$ $=$ $=$	$103$ $129$ $\,\,\,26$
$\varphi (245,\,\,\,4)=\varphi (245,\,\,\,3)-\varphi (\,\,\,35,3)=\,\,\,65-\,\,\,\,\,\,9$  $\varphi (245,\,\,\,3)=\varphi (245,\,\,\,2)-\varphi (\,\,\,49,2)=\,\,\,82-\,\,\,17$  $\varphi (245,\,\,\,2)=\varphi (245,\,\,\,1)-\varphi (\,\,\,81,1)=123-\,\,\,41$  $\varphi (49,\,\,\,2)=\varphi (49,\,\,\,1)-\varphi (\,\,\,16,1)=\,\,\,25-\,\,\,\,\,\,8$	$=$ $=$ $=$ $=$	$\,\,\,56$ $\,\,\,65$ $\,\,\,82$ $\,\,\,17$
$\varphi (\,\,\,35,\,\,\,3)=\varphi (\,\,\,35,\,\,\,2)-\varphi (\,\,\,\,\,\,7,2)=\,\,\,12-\,\,\,\,\,\,3$  $\varphi (\,\,\,35,\,\,\,2)=\varphi (\,\,\,35,\,\,\,1)-\varphi (\,\,\,11,1)=\,\,\,18-\,\,\,\,\,\,6$  $\varphi (7,\,\,\,2)=\varphi (\,\,\,7,\,\,\,1)-\varphi (\,\,\,2,1)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,9$ $\,\,\,12$ $\,\,\,\,\,\,3$
$\varphi (207,\,\,\,5)=\varphi (207,\,\,\,4)-\varphi (\,\,\,18,4)=\,\,\,47-\,\,\,\,\,\,4$  $\varphi (207,\,\,\,4)=\varphi (207,\,\,\,3)-\varphi (\,\,\,29,3)=\,\,\,55-\,\,\,\,\,\,8$  $\varphi (207,\,\,\,3)=\varphi (207,\,\,\,2)-\varphi (\,\,\,41,2)=\,\,\,69-\,\,\,14$  $\varphi (207,\,\,\,2)=\varphi (207,\,\,\,1)-\varphi (\,\,\,69,1)=104-\,\,\,35$  $\varphi (41,\,\,\,2)=\varphi (\,\,\,41,\,\,\,1)-\varphi (\,\,\,13,1)=\,\,\,21-\,\,\,\,\,\,7$	$=$ $=$ $=$ $=$ $=$	$\,\,\,43$ $\,\,\,47$ $\,\,\,55$ $\,\,\,69$ $\,\,\,14$
$\varphi (29,\,\,\,3)=\varphi (\,\,\,29,\,\,\,2)-\varphi (\,\,\,\,\,\,5,2)=\,\,\,10-\,\,\,\,\,\,2$  $\varphi (29,\,\,\,2)=\varphi (\,\,\,29,\,\,\,1)-\varphi (\,\,\,\,\,\,9,2)=\,\,\,15-\,\,\,\,\,\,5$  $\varphi (\,\,\,5,\,\,\,2)=\varphi (\,\,\,\,\,\,5,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,8$ $\,\,\,10$ $\,\,\,\,\,\,2$
$\varphi (18,\,\,\,4)=\varphi (\,\,\,18,\,\,\,3)-\varphi (\,\,\,2,3)=\,\,\,\,\,\,5-\,\,\,\,\,\,1$  $\varphi (18,\,\,\,3)=\varphi (\,\,\,18,\,\,\,2)-\varphi (\,\,\,3,2)=\,\,\,\,\,\,6-\,\,\,\,\,\,1$  $\varphi (18,\,\,\,2)=\varphi (\,\,\,18,\,\,\,1)-\varphi (\,\,\,6,1)=\,\,\,\,\,\,9-\,\,\,\,\,\,3$	$=$ $=$ $=$	$\,\,\,\,\,\,4$ $\,\,\,\,\,\,5$ $\,\,\,\,\,\,6$
$\varphi (158,\,\,\,6)=\varphi (158,\,\,\,5)-\varphi (\,\,\,12,5)=\,\,\,33-\,\,\,\,\,\,1=\,\,\,32$  $\varphi (158,\,\,\,5)=\varphi (158,\,\,\,4)-\varphi (\,\,\,14,4)=\,\,\,36-\,\,\,\,\,\,3=\,\,\,33$  $\varphi (158,\,\,\,4)=\varphi (158,\,\,\,3)-\varphi (\,\,\,22,3)=\,\,\,42-\,\,\,\,\,\,6=\,\,\,36$  [18]$\varphi (158,\,\,\,3)=\varphi (158,\,\,\,2)-\varphi (\,\,\,31,2)=\,\,\,53-\,\,\,11=\,\,\,42$  $\varphi (158,\,\,\,2)=\varphi (158,\,\,\,1)-\varphi (\,\,\,52,1)=\,\,\,79-\,\,\,26=\,\,\,53$  $\varphi (\,\,\,31,\,\,\,2)=\varphi (31,\,\,\,1)-\varphi (10,1)=\,\,\,16-\,\,\,\,\,\,5$	$=$	$\,\,\,11$
$\varphi (\,\,\,22,\,\,\,3)=\varphi (\,\,\,22,\,\,\,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,7-\,\,\,\,\,\,1=\,\,\,\,\,\,6$  $\varphi (\,\,\,22,\,\,\,2)=\varphi (\,\,\,22,\,\,\,1)-\varphi (\,\,\,\,\,\,7,1)=\,\,\,11-\,\,\,\,\,\,4=\,\,\,\,\,\,7$  $\varphi (4,\,\,\,2)=\varphi (\,\,\,4,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,2-\,\,\,\,\,\,1$	$=$	$\,\,\,\,\,\,1$
$\varphi (\,\,\,14,\,\,\,4)=\varphi (\,\,\,14,\,\,\,3)-\varphi (\,\,\,\,\,\,2,3)=\,\,\,\,\,\,4-\,\,\,\,\,\,1=\,\,\,\,\,\,3$  $\varphi (\,\,\,14,\,\,\,3)=\varphi (\,\,\,14,\,\,\,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1=\,\,\,\,\,\,4$  $\varphi (\,\,\,14,\,\,\,2)=\varphi (\,\,\,14,\,\,\,1)-\varphi (\,\,\,\,\,\,4,1)=\,\,\,\,\,\,7-\,\,\,\,\,\,2=\,\,\,\,\,\,5$
$\varphi (142,\,\,\,7)=\varphi (142,\,\,\,6)-\varphi (\,\,\,\,\,\,8,6)=\,\,\,29-\,\,\,\,\,\,1=\,\,\,28$  $\varphi (142,\,\,\,6)=\varphi (142,\,\,\,5)-\varphi (\,\,\,10,5)=\,\,\,30-\,\,\,\,\,\,1=\,\,\,29$  $\varphi (142,\,\,\,5)=\varphi (142,\,\,\,4)-\varphi (\,\,\,12,4)=\,\,\,32-\,\,\,\,\,\,2=\,\,\,30$  $\varphi (142,\,\,\,4)=\varphi (142,\,\,\,3)-\varphi (\,\,\,20,3)=\,\,\,38-\,\,\,\,\,\,6=\,\,\,32$  $\varphi (142,\,\,\,3)=\varphi (142,\,\,\,2)-\varphi (\,\,\,28,2)=\,\,\,47-\,\,\,\,\,\,9=\,\,\,38$  $\varphi (142,\,\,\,2)=\varphi (142,\,\,\,1)-\varphi (\,\,\,47,1)=\,\,\,71-\,\,\,24=\,\,\,47$  $\varphi (\,\,\,28,\,\,\,2)=\varphi (28,\,\,\,1)-\varphi (9,1)=\,\,\,14-\,\,\,\,\,\,5$	$=$	$\,\,\,\,\,\,9$
$\varphi (\,\,\,20,\,\,\,3)=\varphi (\,\,\,20,\,\,\,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,7-\,\,\,\,\,\,1=\,\,\,\,\,\,6$  $\varphi (\,\,\,20,\,\,\,2)=\varphi (\,\,\,20,\,\,\,1)-\varphi (\,\,\,\,\,\,6,1)=\,\,\,10-\,\,\,\,\,\,3=\,\,\,\,\,\,7$  $\varphi (4,\,\,\,2)=\varphi (\,\,\,4,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,\,2-\,\,\,\,\,\,1=\,\,\,\,\,\,1$
$\varphi (\,\,\,12,\,\,\,4)=\varphi (\,\,\,12,\,\,\,3)-\varphi (\,\,\,\,\,\,1,3)=\,\,\,\,\,\,3-\,\,\,\,\,\,1=\,\,\,\,\,\,2$  $\varphi (\,\,\,12,\,\,\,3)=\varphi (\,\,\,12,\,\,\,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,4-\,\,\,\,\,\,1=\,\,\,\,\,\,3$  $\varphi (\,\,\,12,\,\,\,2)=\varphi (\,\,\,12,\,\,\,1)-\varphi (\,\,\,\,\,\,4,1)=\,\,\,\,\,\,6-\,\,\,\,\,\,2=\,\,\,\,\,\,4$
$\varphi (117,\,\,\,8)=\varphi (117,\,\,\,7)-\varphi (\,\,\,6,\,\,\,7)=\,\,\,24-\,\,\,\,\,\,1=\,\,\,23$  $\varphi (117,\,\,\,7)=\varphi (117,\,\,\,6)-\varphi (\,\,\,6,\,\,\,6)=\,\,\,25-\,\,\,\,\,\,1=\,\,\,24$  $\varphi (117,\,\,\,6)=\varphi (117,\,\,\,5)-\varphi (\,\,\,9,\,\,\,5)=\,\,\,26-\,\,\,\,\,\,1=\,\,\,25$  $\varphi (117,\,\,\,5)=\varphi (117,\,\,\,4)-\varphi (10,\,\,\,4)=\,\,\,27-\,\,\,\,\,\,1=\,\,\,26$  $\varphi (117,\,\,\,4)=\varphi (117,\,\,\,3)-\varphi (16,\,\,\,3)=\,\,\,31-\,\,\,\,\,\,4=\,\,\,27$  $\varphi (117,\,\,\,3)=\varphi (117,\,\,\,2)-\varphi (39,\,\,\,1)=\,\,\,59-\,\,\,20=\,\,\,39$  $\varphi (\,\,\,23,\,\,\,2)=\varphi (\,\,\,23,\,\,\,1)-\varphi (\,\,\,\,\,\,7,1)=\,\,\,12-\,\,\,\,\,\,4=\,\,\,\,\,\,8$
$\varphi (\,\,\,16,\,\,\,3)=\varphi (\,\,\,16,\,\,\,2)-\varphi (\,\,\,\,\,\,3,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1=\,\,\,\,\,\,4$  $\varphi (\,\,\,16,\,\,\,2)=\varphi (\,\,\,16,\,\,\,1)-\varphi (\,\,\,\,\,\,5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3=\,\,\,\,\,\,5$
$\varphi (\,\,\,93,\,\,\,9)=\varphi (\,\,\,93,\,\,\,8)-\varphi (\,\,\,\,\,\,4,8)=\,\,\,17-\,\,\,\,\,\,1=\,\,\,16$  $\varphi (\,\,\,93,\,\,\,8)=\varphi (\,\,\,93,\,\,\,7)-\varphi (\,\,\,\,\,\,4,7)=\,\,\,18-\,\,\,\,\,\,1=\,\,\,17$  $\varphi (\,\,\,93,\,\,\,7)=\varphi (\,\,\,93,\,\,\,6)-\varphi (\,\,\,\,\,\,5,6)=\,\,\,19-\,\,\,\,\,\,1=\,\,\,18$  [19]$\varphi (\,\,\,93,\,\,\,6)=\varphi (\,\,\,93,\,\,\,5)-\varphi (\,\,\,\,\,\,7,5)=\,\,\,20-\,\,\,\,\,\,1=\,\,\,19$  $\varphi (\,\,\,93,\,\,\,5)=\varphi (\,\,\,93,\,\,\,4)-\varphi (\,\,\,\,\,\,8,4)=\,\,\,21-\,\,\,\,\,\,1=\,\,\,20$  $\varphi (\,\,\,93,\,\,\,4)=\varphi (\,\,\,93,\,\,\,3)-\varphi (\,\,\,13,3)=\,\,\,25-\,\,\,\,\,\,4=\,\,\,21$  $\varphi (\,\,\,93,\,\,\,3)=\varphi (\,\,\,93,\,\,\,2)-\varphi (\,\,\,18,2)=\,\,\,31-\,\,\,\,\,\,6=\,\,\,25$  $\varphi (\,\,\,93,\,\,\,2)=\varphi (\,\,\,93,\,\,\,1)-\varphi (\,\,\,31,1)=\,\,\,47-\,\,\,16=\,\,\,31$  $\varphi (\,\,\,18,\,\,\,2)=\varphi (18,\,\,\,1)-\varphi (6,1)=\,\,\,\,\,\,9-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,6$
$\varphi (\,\,\,13,\,\,\,3)=\varphi (13,\,\,\,2)-\varphi (2,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1$  $\varphi (\,\,\,13,\,\,\,2)=\varphi (13,\,\,\,1)-\varphi (4,1)=\,\,\,\,\,\,7-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,\,\,\,4$ $\,\,\,\,\,\,5$
$\varphi (\,\,\,87,10)=\varphi (\,\,\,87,\,\,\,9)-\varphi (\,\,\,3,\,\,\,9)=\,\,\,15-\,\,\,\,\,\,1=\,\,\,14$  $\varphi (\,\,\,87,\,\,\,9)=\varphi (\,\,\,87,\,\,\,8)-\varphi (\,\,\,3,\,\,\,8)=\,\,\,16-\,\,\,\,\,\,1=\,\,\,15$  $\varphi (\,\,\,87,\,\,\,8)=\varphi (\,\,\,87,\,\,\,7)-\varphi (\,\,\,4,\,\,\,7)=\,\,\,17-\,\,\,\,\,\,1=\,\,\,16$  $\varphi (\,\,\,87,\,\,\,7)=\varphi (\,\,\,87,\,\,\,6)-\varphi (\,\,\,5,\,\,\,6)=\,\,\,18-\,\,\,\,\,\,1=\,\,\,17$  $\varphi (\,\,\,87,\,\,\,6)=\varphi (\,\,\,87,\,\,\,5)-\varphi (\,\,\,6,\,\,\,5)=\,\,\,19-\,\,\,\,\,\,1=\,\,\,18$  $\varphi (\,\,\,87,\,\,\,5)=\varphi (\,\,\,87,\,\,\,4)-\varphi (\,\,\,7,\,\,\,4)=\,\,\,20-\,\,\,\,\,\,1=\,\,\,19$  $\varphi (\,\,\,87,\,\,\,4)=\varphi (\,\,\,87,\,\,\,3)-\varphi (12,\,\,\,3)=\,\,\,23-\,\,\,\,\,\,3=\,\,\,20$  $\varphi (\,\,\,87,\,\,\,3)=\varphi (\,\,\,87,\,\,\,2)-\varphi (17,\,\,\,2)=\,\,\,29-\,\,\,\,\,\,6=\,\,\,23$  $\varphi (\,\,\,87,\,\,\,2)=\varphi (\,\,\,87,\,\,\,1)-\varphi (29,\,\,\,1)=\,\,\,44-\,\,\,15=\,\,\,29$  $\varphi (\,\,\,17,\,\,\,2)=\varphi (17,\,\,\,1)-\varphi (5,1)=\,\,\,\,\,\,9-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,6$
$\varphi (\,\,\,12,\,\,\,3)=\varphi (12,\,\,\,2)-\varphi (2,2)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$  $\varphi (\,\,\,12,\,\,\,2)=\varphi (12,\,\,\,1)-\varphi (4,1)=\,\,\,\,\,\,6-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,\,\,\,3$ $\,\,\,\,\,\,4$
$\varphi (2439,12)=\varphi (2439,11)-\varphi (\,\,\,65,11)=\,\,\,367-\,\,\,\,\,\,8$  $\varphi (2439,11)=\varphi (2439,10)-\varphi (\,\,\,78,10)=\,\,\,379-\,\,\,12$  $\varphi (2439,10)=\varphi (2439,\,\,\,9)-\varphi (\,\,\,84,\,\,\,9)=\,\,\,394-\,\,\,15$  $\varphi (2439,\,\,\,9)=\varphi (2439,\,\,\,8)-\varphi (106,\,\,\,8)=\,\,\,414-\,\,\,20$  $\varphi (2439,\,\,\,8)=\varphi (2439,\,\,\,7)-\varphi (128,\,\,\,7)=\,\,\,439-\,\,\,25$  $\varphi (2439,\,\,\,7)=\varphi (2439,\,\,\,6)-\varphi (143,\,\,\,6)=\,\,\,468-\,\,\,29$  $\varphi (2439,\,\,\,6)=\varphi (2439,\,\,\,5)-\varphi (187,\,\,\,5)=\,\,\,507-\,\,\,39$  $\varphi (2439,\,\,\,5)=\varphi (2439,\,\,\,4)-\varphi (221,\,\,\,4)=\,\,\,557-\,\,\,50$  $\varphi (2439,\,\,\,4)=\varphi (2439,\,\,\,3)-\varphi (348,\,\,\,3)=\,\,\,650-\,\,\,93$  $\varphi (2439,\,\,\,3)=\varphi (2439,\,\,\,2)-\varphi (487,\,\,\,2)=\,\,\,813-163$  $\varphi (2439,\,\,\,2)=\varphi (2439,\,\,\,1)-\varphi (813,\,\,\,1)=1220-407$  $\varphi (487,\,\,\,2)=\varphi (487,\,\,\,1)-\varphi (162,1)=244-\,\,\,81$	$=$ $=$ $=$ $=$ $=$ $=$ $=$ $=$ $=$ $=$ $=$ $=$	$359$ $367$ $379$ $394$ $414$ $439$ $468$ $507$ $557$ $650$ $813$ $163$
$\varphi (\,\,\,348,\,\,\,3)=\varphi (348,\,\,\,2)-\varphi (\,\,\,69,2)=116-\,\,\,23$  $\varphi (\,\,\,348,\,\,\,2)=\varphi (348,\,\,\,1)-\varphi (116,1)=174-\,\,\,58$  $\varphi (69,\,\,\,2)=\varphi (69,\,\,\,1)-\varphi (\,\,\,23,1)=\,\,\,35-\,\,\,12$	$=$ $=$ $=$	$\,\,\,93$ $116$ $\,\,\,23$
$\varphi (221,\,\,\,4)=\varphi (221,\,\,\,3)-\varphi (31,\,\,\,3)=\,\,\,59-\,\,\,\,\,\,9=\,\,\,50$  $\varphi (221,\,\,\,3)=\varphi (221,\,\,\,2)-\varphi (42,\,\,\,2)=\,\,\,74-\,\,\,15=\,\,\,59$  $\varphi (221,\,\,\,2)=\varphi (221,\,\,\,1)-\varphi (73,\,\,\,1)=111-\,\,\,37=\,\,\,74$  $\varphi (\,\,\,44,\,\,\,2)=\varphi (44,\,\,\,1)-\varphi (14,1)=\,\,\,22-\,\,\,\,\,\,7=\,\,\,15$
$\varphi (31,\,\,\,3)=\varphi (31,\,\,\,2)-\varphi (\,\,\,6,2)=\,\,\,11-\,\,\,\,\,\,2=\,\,\,\,\,\,9$  $\varphi (31,\,\,\,2)=\varphi (31,\,\,\,1)-\varphi (10,1)=\,\,\,16-\,\,\,\,\,\,5=\,\,\,11$  $\varphi (6,\,\,\,2)=\varphi (6,\,\,\,1)-\varphi (\,\,\,2,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1=\,\,\,\,\,\,2$
$\varphi (187,\,\,\,5)=\varphi (187,\,\,\,4)-\varphi (17,\,\,\,4)=\,\,\,43-\,\,\,\,\,\,4=\,\,\,39$  $\varphi (187,\,\,\,4)=\varphi (187,\,\,\,3)-\varphi (26,\,\,\,3)=\,\,\,50-\,\,\,\,\,\,7=\,\,\,43$  $\varphi (187,\,\,\,3)=\varphi (187,\,\,\,2)-\varphi (37,\,\,\,2)=\,\,\,63-\,\,\,13=\,\,\,50$  $\varphi (187,\,\,\,2)=\varphi (187,\,\,\,1)-\varphi (62,\,\,\,1)=\,\,\,94-\,\,\,31=\,\,\,63$  $\varphi (37,\,\,\,2)=\varphi (37,\,\,\,1)-\varphi (12,1)=\,\,\,19-\,\,\,\,\,\,6=\,\,\,13$
$\varphi (26,\,\,\,3)=\varphi (26,\,\,\,2)-\varphi (5,\,\,\,2)=\,\,\,\,\,\,9-\,\,\,\,\,\,2=\,\,\,\,\,\,7$  $\varphi (26,\,\,\,2)=\varphi (26,\,\,\,1)-\varphi (8,\,\,\,1)=\,\,\,13-\,\,\,\,\,\,4=\,\,\,\,\,\,9$  $\varphi (5,\,\,\,2)=\varphi (\,\,\,5,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1=\,\,\,\,\,\,2$
$\varphi (17,\,\,\,4)=\varphi (17,\,\,\,3)-\varphi (2,\,\,\,3)=\,\,\,\,\,\,5-\,\,\,\,\,\,1=\,\,\,\,\,\,4$  $\varphi (17,\,\,\,3)=\varphi (17,\,\,\,2)-\varphi (3,\,\,\,2)=\,\,\,\,\,\,6-\,\,\,\,\,\,1=\,\,\,\,\,\,5$  $\varphi (17,\,\,\,2)=\varphi (17,\,\,\,1)-\varphi (5,\,\,\,1)=\,\,\,\,\,\,9-\,\,\,\,\,\,3=\,\,\,\,\,\,6$
$\varphi (143,\,\,\,6)=\varphi (143,\,\,\,5)-\varphi (11,5)=\,\,\,30-\,\,\,\,\,\,1=\,\,\,29$  $\varphi (143,\,\,\,5)=\varphi (143,\,\,\,4)-\varphi (13,4)=\,\,\,33-\,\,\,\,\,\,3=\,\,\,30$  $\varphi (143,\,\,\,4)=\varphi (143,\,\,\,3)-\varphi (20,3)=\,\,\,39-\,\,\,\,\,\,6=\,\,\,33$  $\varphi (143,\,\,\,3)=\varphi (143,\,\,\,2)-\varphi (28,2)=\,\,\,48-\,\,\,\,\,\,9=\,\,\,39$  $\varphi (143,\,\,\,2)=\varphi (143,\,\,\,1)-\varphi (47,1)=\,\,\,72-\,\,\,24=\,\,\,48$  $\varphi (28,\,\,\,2)=\varphi (28,\,\,\,1)-\varphi (9,1)=\,\,\,14-\,\,\,\,\,\,5=\,\,\,9$
$\varphi (20,\,\,\,3)=\varphi (20,\,\,\,2)-\varphi (4,2)=\,\,\,\,\,\,7-\,\,\,\,\,\,1=\,\,\,\,\,\,6$  $\varphi (20,\,\,\,2)=\varphi (20,\,\,\,1)-\varphi (6,1)=\,\,\,10-\,\,\,\,\,\,3=\,\,\,\,\,\,7$  $\varphi (4,\,\,\,2)=\varphi (\,\,\,4,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,\,2-\,\,\,\,\,\,1=\,\,\,\,\,\,1$
$\varphi (13,\,\,\,4)=\varphi (13,\,\,\,3)-\varphi (1,\,\,\,3)=\,\,\,\,\,\,4-\,\,\,\,\,\,1=\,\,\,\,\,\,3$  $\varphi (13,\,\,\,3)=\varphi (13,\,\,\,2)-\varphi (2,\,\,\,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1=\,\,\,\,\,\,4$  $\varphi (13,\,\,\,2)=\varphi (13,\,\,\,1)-\varphi (4,\,\,\,1)=\,\,\,\,\,\,7-\,\,\,\,\,\,2=\,\,\,\,\,\,5$
$\varphi (128,\,\,\,7)=\varphi (128,\,\,\,6)-\varphi (\,\,\,7,\,\,\,6)=\,\,\,26-\,\,\,\,\,\,1=\,\,\,25$  $\varphi (128,\,\,\,6)=\varphi (128,\,\,\,5)-\varphi (\,\,\,9,\,\,\,5)=\,\,\,27-\,\,\,\,\,\,1=\,\,\,26$  $\varphi (128,\,\,\,5)=\varphi (128,\,\,\,4)-\varphi (11,\,\,\,4)=\,\,\,29-\,\,\,\,\,\,2=\,\,\,27$  $\varphi (128,\,\,\,4)=\varphi (128,\,\,\,3)-\varphi (18,\,\,\,3)=\,\,\,34-\,\,\,\,\,\,5=\,\,\,29$  [21]$\varphi (128,\,\,\,3)=\varphi (128,\,\,\,2)-\varphi (25,\,\,\,2)=\,\,\,43-\,\,\,\,\,\,9=\,\,\,34$  $\varphi (128,\,\,\,2)=\varphi (128,\,\,\,1)-\varphi (42,\,\,\,1)=\,\,\,62-\,\,\,21=\,\,\,43$  $\varphi (\,\,\,25,\,\,\,2)=\varphi (25,\,\,\,1)-\varphi (8,1)=\,\,\,13-\,\,\,\,\,\,4$	$=$	$\,\,\,\,\,\,9$
$\varphi (\,\,\,18,\,\,\,3)=\varphi (18,\,\,\,2)-\varphi (3,2)=\,\,\,\,\,\,6-\,\,\,\,\,\,1$  $\varphi (\,\,\,18,\,\,\,2)=\varphi (18,\,\,\,1)-\varphi (6,1)=\,\,\,\,\,\,9-\,\,\,\,\,\,3$	$=$ $=$	$\,\,\,\,\,\,5$ $\,\,\,\,\,\,6$
$\varphi (\,\,\,11,\,\,\,4)=\varphi (11,\,\,\,3)-\varphi (1,3)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$  $\varphi (\,\,\,11,\,\,\,3)=\varphi (11,\,\,\,2)-\varphi (2,2)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$  $\varphi (\,\,\,11,\,\,\,2)=\varphi (11,\,\,\,1)-\varphi (3,1)=\,\,\,\,\,\,6-\,\,\,\,\,\,2$	$=$ $=$ $=$	$\,\,\,\,\,\,2$ $\,\,\,\,\,\,3$ $\,\,\,\,\,\,4$
$\varphi (106,\,\,\,8)=\varphi (106,\,\,\,7)-\varphi (\,\,\,5,\,\,\,7)=\,\,\,21-\,\,\,\,\,\,1=\,\,\,20$  $\varphi (106,\,\,\,7)=\varphi (106,\,\,\,6)-\varphi (\,\,\,6,\,\,\,6)=\,\,\,22-\,\,\,\,\,\,1=\,\,\,21$  $\varphi (106,\,\,\,6)=\varphi (106,\,\,\,5)-\varphi (\,\,\,8,\,\,\,5)=\,\,\,23-\,\,\,\,\,\,1=\,\,\,22$  $\varphi (106,\,\,\,5)=\varphi (106,\,\,\,4)-\varphi (\,\,\,9,\,\,\,4)=\,\,\,24-\,\,\,\,\,\,1=\,\,\,23$  $\varphi (106,\,\,\,4)=\varphi (106,\,\,\,3)-\varphi (15,\,\,\,3)=\,\,\,28-\,\,\,\,\,\,4=\,\,\,24$  $\varphi (106,\,\,\,3)=\varphi (106,\,\,\,2)-\varphi (21,\,\,\,2)=\,\,\,35-\,\,\,\,\,\,7=\,\,\,28$  $\varphi (106,\,\,\,2)=\varphi (106,\,\,\,1)-\varphi (35,\,\,\,1)=\,\,\,53-\,\,\,18=\,\,\,35$  $\varphi (\,\,\,21,\,\,\,2)=\varphi (21,\,\,\,1)-\varphi (7,1)=\,\,\,11-\,\,\,\,\,\,4$	$=$	$\,\,\,\,\,\,7$
$\varphi (\,\,\,15,\,\,\,3)=\varphi (15,\,\,\,2)-\varphi (3,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1$  $\varphi (\,\,\,15,\,\,\,2)=\varphi (15,\,\,\,1)-\varphi (5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3$	$=$ $=$	$\,\,\,\,\,\,4$ $\,\,\,\,\,\,5$
$\varphi (\,\,\,84,\,\,\,9)=\varphi (\,\,\,84,\,\,\,8)-\varphi (\,\,\,3,\,\,\,8)=\,\,\,16-\,\,\,\,\,\,1=\,\,\,15$  $\varphi (\,\,\,84,\,\,\,8)=\varphi (\,\,\,84,\,\,\,7)-\varphi (\,\,\,4,\,\,\,7)=\,\,\,17-\,\,\,\,\,\,1=\,\,\,16$  $\varphi (\,\,\,84,\,\,\,7)=\varphi (\,\,\,84,\,\,\,6)-\varphi (\,\,\,4,\,\,\,6)=\,\,\,18-\,\,\,\,\,\,1=\,\,\,17$  $\varphi (\,\,\,84,\,\,\,6)=\varphi (\,\,\,84,\,\,\,5)-\varphi (\,\,\,6,\,\,\,5)=\,\,\,19-\,\,\,\,\,\,1=\,\,\,18$  $\varphi (\,\,\,84,\,\,\,5)=\varphi (\,\,\,84,\,\,\,4)-\varphi (\,\,\,7,\,\,\,4)=\,\,\,20-\,\,\,\,\,\,1=\,\,\,19$  $\varphi (\,\,\,84,\,\,\,4)=\varphi (\,\,\,84,\,\,\,3)-\varphi (12,\,\,\,3)=\,\,\,23-\,\,\,\,\,\,3=\,\,\,20$  $\varphi (\,\,\,84,\,\,\,3)=\varphi (\,\,\,84,\,\,\,2)-\varphi (16,\,\,\,2)=\,\,\,28-\,\,\,\,\,\,5=\,\,\,23$  $\varphi (\,\,\,84,\,\,\,2)=\varphi (\,\,\,84,\,\,\,1)-\varphi (28,\,\,\,1)=\,\,\,42-\,\,\,14=\,\,\,28$  $\varphi (\,\,\,16,\,\,\,2)=\varphi (16,\,\,\,1)-\varphi (5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,5$
$\varphi (\,\,\,12,3)=\varphi (12,2)-\varphi (2,2)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$  $\varphi (\,\,\,12,2)=\varphi (12,1)-\varphi (4,1)=\,\,\,\,\,\,6-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,\,\,\,3$ $\,\,\,\,\,\,4$
$\varphi (\,\,\,78,10)=\varphi (\,\,\,78,\,\,\,9)-\varphi (\,\,\,2,\,\,\,9)=\,\,\,13-\,\,\,\,\,\,1=\,\,\,12$  $\varphi (\,\,\,78,\,\,\,9)=\varphi (\,\,\,78,\,\,\,8)-\varphi (\,\,\,3,\,\,\,8)=\,\,\,14-\,\,\,\,\,\,1=\,\,\,13$  $\varphi (\,\,\,78,\,\,\,8)=\varphi (\,\,\,78,\,\,\,7)-\varphi (\,\,\,4,\,\,\,7)=\,\,\,15-\,\,\,\,\,\,1=\,\,\,14$  $\varphi (\,\,\,78,\,\,\,7)=\varphi (\,\,\,78,\,\,\,6)-\varphi (\,\,\,4,\,\,\,6)=\,\,\,16-\,\,\,\,\,\,1=\,\,\,15$  $\varphi (\,\,\,78,\,\,\,6)=\varphi (\,\,\,78,\,\,\,5)-\varphi (\,\,\,6,\,\,\,5)=\,\,\,17-\,\,\,\,\,\,1=\,\,\,16$  $\varphi (\,\,\,78,\,\,\,5)=\varphi (\,\,\,78,\,\,\,4)-\varphi (\,\,\,7,\,\,\,4)=\,\,\,18-\,\,\,\,\,\,1=\,\,\,17$  [22]$\varphi (\,\,\,78,\,\,\,4)=\varphi (\,\,\,78,\,\,\,3)-\varphi (\,\,\,11,\,\,\,3)=\,\,\,21-\,\,\,\,\,\,3=\,\,\,18$  $\varphi (\,\,\,78,\,\,\,3)=\varphi (\,\,\,78,\,\,\,2)-\varphi (\,\,\,15,\,\,\,2)=\,\,\,26-\,\,\,\,\,\,5=\,\,\,21$  $\varphi (\,\,\,78,\,\,\,2)=\varphi (\,\,\,78,\,\,\,1)-\varphi (\,\,\,26,\,\,\,1)=\,\,\,39-\,\,\,13=\,\,\,26$  $\varphi (\,\,\,15,\,\,\,2)=\varphi (15,\,\,\,1)-\varphi (5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,5$
$\varphi (\,\,\,65,11)=\varphi (\,\,\,65,10)-\varphi (\,\,\,2,10)=\,\,\,\,\,\,9-\,\,\,\,\,\,1=\,\,\,\,\,\,8$  $\varphi (\,\,\,65,10)=\varphi (\,\,\,65,\,\,\,9)-\varphi (\,\,\,2,\,\,\,9)=\,\,\,10-\,\,\,\,\,\,1=\,\,\,\,\,\,9$  $\varphi (\,\,\,65,\,\,\,9)=\varphi (\,\,\,65,\,\,\,8)-\varphi (\,\,\,2,\,\,\,8)=\,\,\,11-\,\,\,\,\,\,1=\,\,\,10$  $\varphi (\,\,\,65,\,\,\,8)=\varphi (\,\,\,65,\,\,\,7)-\varphi (\,\,\,3,\,\,\,7)=\,\,\,12-\,\,\,\,\,\,1=\,\,\,11$  $\varphi (\,\,\,65,\,\,\,7)=\varphi (\,\,\,65,\,\,\,6)-\varphi (\,\,\,3,\,\,\,6)=\,\,\,13-\,\,\,\,\,\,1=\,\,\,12$  $\varphi (\,\,\,65,\,\,\,6)=\varphi (\,\,\,65,\,\,\,5)-\varphi (\,\,\,5,\,\,\,5)=\,\,\,14-\,\,\,\,\,\,1=\,\,\,13$  $\varphi (\,\,\,65,\,\,\,5)=\varphi (\,\,\,65,\,\,\,4)-\varphi (\,\,\,5,\,\,\,4)=\,\,\,15-\,\,\,\,\,\,1=\,\,\,14$  $\varphi (\,\,\,65,\,\,\,4)=\varphi (\,\,\,65,\,\,\,3)-\varphi (\,\,\,9,\,\,\,3)=\,\,\,17-\,\,\,\,\,\,2=\,\,\,15$  $\varphi (\,\,\,65,\,\,\,3)=\varphi (\,\,\,65,\,\,\,2)-\varphi (13,\,\,\,2)=\,\,\,22-\,\,\,\,\,\,5=\,\,\,17$  $\varphi (\,\,\,65,\,\,\,2)=\varphi (\,\,\,65,\,\,\,1)-\varphi (21,\,\,\,1)=\,\,\,33-\,\,\,11=\,\,\,22$  $\varphi (\,\,\,13,\,\,\,2)=\varphi (13,\,\,\,1)-\varphi (4,1)=\,\,\,\,\,\,7-\,\,\,\,\,\,2$	$=$	$\,\,\,\,\,\,5$
$\varphi (\,\,\,9,\,\,\,3)=\varphi (\,\,\,9,\,\,\,2)-\varphi (1,2)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$  $\varphi (\,\,\,9,\,\,\,2)=\varphi (\,\,\,9,\,\,\,1)-\varphi (3,1)=\,\,\,\,\,\,5-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,\,\,\,2$ $\,\,\,\,\,\,3$
$\varphi (2325,13)=\varphi (2325,12)-\varphi (\,\,\,56,12)=\,\,\,341-\,\,\,\,\,\,5=336$  $\varphi (2325,12)=\varphi (2325,11)-\varphi (\,\,\,62,11)=\,\,\,349-\,\,\,\,\,\,8=341$  $\varphi (2325,11)=\varphi (2325,10)-\varphi (\,\,\,78,10)=\,\,\,361-\,\,\,12=349$  $\varphi (2325,10)=\varphi (2325,\,\,\,9)-\varphi (\,\,\,80,\,\,\,9)=\,\,\,375-\,\,\,14=361$  $\varphi (2325,\,\,\,9)=\varphi (2325,\,\,\,8)-\varphi (101,\,\,\,8)=\,\,\,394-\,\,\,19=375$  $\varphi (2325,\,\,\,8)=\varphi (2325,\,\,\,7)-\varphi (122,\,\,\,7)=\,\,\,418-\,\,\,24=394$  $\varphi (2325,\,\,\,7)=\varphi (2325,\,\,\,6)-\varphi (136,\,\,\,6)=\,\,\,445-\,\,\,27=418$  $\varphi (2325,\,\,\,6)=\varphi (2325,\,\,\,5)-\varphi (178,\,\,\,5)=\,\,\,482-\,\,\,37=445$  $\varphi (2325,\,\,\,5)=\varphi (2325,\,\,\,4)-\varphi (211,\,\,\,4)=\,\,\,531-\,\,\,49=482$  $\varphi (2325,\,\,\,4)=\varphi (2325,\,\,\,3)-\varphi (332,\,\,\,3)=\,\,\,620-\,\,\,89=531$  $\varphi (2325,\,\,\,3)=\varphi (2325,\,\,\,2)-\varphi (465,\,\,\,2)=\,\,\,775-155=620$  $\varphi (2325,\,\,\,2)=\varphi (2325,\,\,\,1)-\varphi (775,\,\,\,1)=1163-388=775$  $\varphi (\,\,\,465,\,\,\,2)=\varphi (465,\,\,\,1)-\varphi (155,1)=233-\,\,\,78$	$=$	$155$
$\varphi (\,\,\,332,\,\,\,3)=\varphi (\,\,\,332,\,\,\,1)-\varphi (\,\,\,66,2)=111-\,\,\,22$  $\varphi (\,\,\,332,\,\,\,2)=\varphi (\,\,\,332,\,\,\,1)-\varphi (110,1)=166-\,\,\,55$  $\varphi (66,\,\,\,2)=\varphi (\,\,\,66,\,\,\,1)-\varphi (\,\,\,22,1)=\,\,\,33-\,\,\,11$	$=$ $=$ $=$	$\,\,\,89$ $111$ $\,\,\,22$
$\varphi (211,\,\,\,4)=\varphi (211,\,\,\,3)-\varphi (30,\,\,\,3)=\,\,\,57-\,\,\,\,\,\,8=\,\,\,49$  $\varphi (211,\,\,\,3)=\varphi (211,\,\,\,2)-\varphi (42,\,\,\,2)=\,\,\,71-\,\,\,14=\,\,\,57$  $\varphi (211,\,\,\,2)=\varphi (211,\,\,\,1)-\varphi (70,\,\,\,1)=106-\,\,\,35=\,\,\,71$  $\varphi (\,\,\,42,\,\,\,2)=\varphi (42,\,\,\,1)-\varphi (14,1)=\,\,\,21-\,\,\,\,\,\,7$	$=$	$\,\,\,14$
$\varphi (\,\,\,30,\,\,\,3)=\varphi (\,\,\,30,\,\,\,2)-\varphi (\,\,\,6,2)=\,\,\,10-\,\,\,\,\,\,2$  $\varphi (\,\,\,30,\,\,\,2)=\varphi (\,\,\,30,\,\,\,1)-\varphi (10,1)=\,\,\,15-\,\,\,\,\,\,5$  $\varphi (\,\,\,6,\,\,\,2)=\varphi (\,\,\,6,\,\,\,1)-\varphi (2,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,8$ $\,\,\,10$ $\,\,\,\,\,\,2$
$\varphi (178,\,\,\,5)=\varphi (178,\,\,\,4)-\varphi (16,\,\,\,4)=\,\,\,40-\,\,\,\,\,\,3=\,\,\,37$  $\varphi (178,\,\,\,4)=\varphi (178,\,\,\,3)-\varphi (25,\,\,\,3)=\,\,\,47-\,\,\,\,\,\,7=\,\,\,40$  $\varphi (178,\,\,\,3)=\varphi (178,\,\,\,2)-\varphi (35,\,\,\,2)=\,\,\,59-\,\,\,12=\,\,\,47$  $\varphi (178,\,\,\,2)=\varphi (178,\,\,\,1)-\varphi (59,\,\,\,1)=\,\,\,89-\,\,\,30=\,\,\,59$  $\varphi (\,\,\,35,\,\,\,2)=\varphi (35,\,\,\,1)-\varphi (11,1)=\,\,\,18-\,\,\,\,\,\,6$	$=$	$\,\,\,12$
$\varphi (\,\,\,25,\,\,\,3)=\varphi (\,\,\,25,\,\,\,2)-\varphi (\,\,\,5,2)=\,\,\,\,\,\,9-\,\,\,\,\,\,2$  $\varphi (\,\,\,25,\,\,\,2)=\varphi (\,\,\,25,\,\,\,1)-\varphi (\,\,\,8,1)=\,\,\,\,13-\,\,\,\,\,\,2$  $\varphi (\,\,\,5,\,\,\,2)=\varphi (\,\,\,5,\,\,\,1)-\varphi (\,\,\,1,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,7$ $\,\,\,\,\,\,9$ $\,\,\,\,\,\,2$
$\varphi (\,\,\,16,\,\,\,4)=\varphi (\,\,\,16,\,\,\,3)-\varphi (\,\,\,2,3)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$  $\varphi (\,\,\,16,\,\,\,3)=\varphi (\,\,\,16,\,\,\,2)-\varphi (\,\,\,3,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1$  $\varphi (\,\,\,16,\,\,\,2)=\varphi (\,\,\,16,\,\,\,1)-\varphi (\,\,\,5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3$	$=$ $=$ $=$	$\,\,\,\,\,\,3$ $\,\,\,\,\,\,4$ $\,\,\,\,\,\,5$
$\varphi (136,\,\,\,6)=\varphi (136,\,\,\,5)-\varphi (10,\,\,\,5)=\,\,\,28-\,\,\,\,\,\,1=\,\,\,27$  $\varphi (136,\,\,\,5)=\varphi (136,\,\,\,4)-\varphi (12,\,\,\,4)=\,\,\,30-\,\,\,\,\,\,2=\,\,\,28$  $\varphi (136,\,\,\,4)=\varphi (136,\,\,\,3)-\varphi (19,\,\,\,3)=\,\,\,36-\,\,\,\,\,\,6=\,\,\,30$  $\varphi (136,\,\,\,3)=\varphi (136,\,\,\,2)-\varphi (27,\,\,\,2)=\,\,\,45-\,\,\,\,\,\,9=\,\,\,36$  $\varphi (136,\,\,\,2)=\varphi (136,\,\,\,1)-\varphi (45,\,\,\,1)=\,\,\,68-\,\,\,23=\,\,\,45$  $\varphi (27,2)=\varphi (27,1)-\varphi (9,\,\,\,1)=\,\,\,14-\,\,\,\,\,\,5$  $\varphi (19,3)=\varphi (19,2)-\varphi (3,\,\,\,2)=\,\,\,\,\,\,7-\,\,\,\,\,\,1$  $\varphi (19,2)=\varphi (19,1)-\varphi (6,\,\,\,1)=\,\,\,10-\,\,\,\,\,\,3$	$=$ $=$ $=$	$\,\,\,\,\,\,9$ $\,\,\,\,\,\,6$ $\,\,\,\,\,\,7$
$\varphi (12,4)=\varphi (12,3)-\varphi (1,\,\,\,3)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$  $\varphi (12,3)=\varphi (12,2)-\varphi (2,\,\,\,2)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$  $\varphi (12,2)=\varphi (12,1)-\varphi (4,\,\,\,1)=\,\,\,\,\,\,6-\,\,\,\,\,\,3$	$=$ $=$ $=$	$\,\,\,\,\,\,2$ $\,\,\,\,\,\,3$ $\,\,\,\,\,\,4$
$\varphi (122,\,\,\,7)=\varphi (136,\,\,\,6)-\varphi (\,\,\,7,\,\,\,6)=\,\,\,25-\,\,\,\,\,\,1=\,\,\,24$  $\varphi (122,\,\,\,6)=\varphi (136,\,\,\,5)-\varphi (\,\,\,9,\,\,\,5)=\,\,\,26-\,\,\,\,\,\,1=\,\,\,25$  $\varphi (122,\,\,\,5)=\varphi (136,\,\,\,4)-\varphi (11,\,\,\,4)=\,\,\,28-\,\,\,\,\,\,2=\,\,\,26$  $\varphi (122,\,\,\,4)=\varphi (136,\,\,\,3)-\varphi (17,\,\,\,3)=\,\,\,33-\,\,\,\,\,\,5=\,\,\,28$  $\varphi (122,\,\,\,3)=\varphi (136,\,\,\,2)-\varphi (24,\,\,\,2)=\,\,\,41-\,\,\,\,\,\,8=\,\,\,33$  $\varphi (122,\,\,\,2)=\varphi (136,\,\,\,1)-\varphi (40,\,\,\,1)=\,\,\,61-\,\,\,20=\,\,\,41$  $\varphi (24,2)=\varphi (24,\,\,\,1)-\varphi (8,1)=\,\,\,12-\,\,\,\,\,\,4$	$=$	$\,\,\,\,\,\,8$
$\varphi (\,\,\,17,\,\,\,3)=\varphi (\,\,\,17,\,\,\,2)-\varphi (3,2)=\,\,\,=\,\,\,6-\,\,\,\,\,\,1=\,\,\,\,\,\,5$  $\varphi (\,\,\,17,\,\,\,2)=\varphi (\,\,\,17,\,\,\,1)-\varphi (5,1)=\,\,\,=\,\,\,9-\,\,\,\,\,\,3=\,\,\,\,\,\,6$
$\varphi (\,\,\,11,\,\,\,4)=\varphi (\,\,\,11,\,\,\,3)-\varphi (1,3)=\,\,\,=\,\,\,3-\,\,\,\,\,\,1=\,\,\,\,\,\,2$  $\varphi (\,\,\,11,\,\,\,3)=\varphi (\,\,\,11,\,\,\,2)-\varphi (2,2)=\,\,\,=\,\,\,4-\,\,\,\,\,\,1=\,\,\,\,\,\,3$  $\varphi (\,\,\,11,\,\,\,2)=\varphi (\,\,\,11,\,\,\,1)-\varphi (3,1)=\,\,\,=\,\,\,6-\,\,\,\,\,\,2=\,\,\,\,\,\,4$
$\varphi (101,\,\,\,8)=\varphi (101,\,\,\,7)-\varphi (\,\,\,5,\,\,\,7)=\,\,\,20-\,\,\,\,\,\,1=\,\,\,19$  $\varphi (101,\,\,\,7)=\varphi (101,\,\,\,6)-\varphi (\,\,\,5,\,\,\,6)=\,\,\,21-\,\,\,\,\,\,1=\,\,\,20$  $\varphi (101,\,\,\,6)=\varphi (101,\,\,\,5)-\varphi (\,\,\,7,\,\,\,5)=\,\,\,22-\,\,\,\,\,\,1=\,\,\,21$  $\varphi (101,\,\,\,5)=\varphi (101,\,\,\,4)-\varphi (\,\,\,9,\,\,\,4)=\,\,\,23-\,\,\,\,\,\,1=\,\,\,22$  $\varphi (101,\,\,\,4)=\varphi (101,\,\,\,3)-\varphi (14,\,\,\,3)=\,\,\,27-\,\,\,\,\,\,4=\,\,\,23$  $\varphi (101,\,\,\,3)=\varphi (101,\,\,\,2)-\varphi (20,\,\,\,2)=\,\,\,34-\,\,\,\,\,\,7=\,\,\,27$  $\varphi (101,\,\,\,2)=\varphi (101,\,\,\,1)-\varphi (33,\,\,\,1)=\,\,\,51-\,\,\,17=\,\,\,34$  $\varphi (\,\,\,20,\,\,\,2)=\varphi (20,\,\,\,1)-\varphi (6,1)=\,\,\,10-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,7$
$\varphi (\,\,\,14,\,\,\,3)=\varphi (14,\,\,\,2)-\varphi (2,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1$  $\varphi (\,\,\,14,\,\,\,2)=\varphi (14,\,\,\,1)-\varphi (4,1)=\,\,\,\,\,\,7-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,\,\,\,4$ $\,\,\,\,\,\,5$
$\varphi (\,\,\,80,\,\,\,9)=\varphi (\,\,\,80,\,\,\,8)-\varphi (\,\,\,3,\,\,\,8)=\,\,\,15-\,\,\,\,\,\,1=\,\,\,14$  $\varphi (\,\,\,80,\,\,\,8)=\varphi (\,\,\,80,\,\,\,7)-\varphi (\,\,\,4,\,\,\,7)=\,\,\,16-\,\,\,\,\,\,1=\,\,\,15$  $\varphi (\,\,\,80,\,\,\,7)=\varphi (\,\,\,80,\,\,\,6)-\varphi (\,\,\,4,\,\,\,6)=\,\,\,17-\,\,\,\,\,\,1=\,\,\,16$  $\varphi (\,\,\,80,\,\,\,5)=\varphi (\,\,\,80,\,\,\,5)-\varphi (\,\,\,6,\,\,\,5)=\,\,\,18-\,\,\,\,\,\,1=\,\,\,17$  $\varphi (\,\,\,80,\,\,\,4)=\varphi (\,\,\,80,\,\,\,3)-\varphi (11,\,\,\,3)=\,\,\,22-\,\,\,\,\,\,3=\,\,\,19$  $\varphi (\,\,\,80,\,\,\,3)=\varphi (\,\,\,80,\,\,\,2)-\varphi (16,\,\,\,2)=\,\,\,27-\,\,\,\,\,\,5=\,\,\,22$  $\varphi (\,\,\,80,\,\,\,2)=\varphi (\,\,\,80,\,\,\,1)-\varphi (26,\,\,\,1)=\,\,\,40-\,\,\,13=\,\,\,27$  $\varphi (\,\,\,16,\,\,\,2)=\varphi (16,\,\,\,1)-\varphi (5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,5$
$\varphi (\,\,\,11,\,\,\,3)=\varphi (11,\,\,\,2)-\varphi (2,2)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$  $\varphi (\,\,\,11,\,\,\,2)=\varphi (11,\,\,\,1)-\varphi (3,1)=\,\,\,\,\,\,6-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,\,\,\,3$ $\,\,\,\,\,\,4$
$\varphi (\,\,\,75,10)=\varphi (\,\,\,75,\,\,\,9)-\varphi (\,\,\,2,9)=\,\,\,=\,\,\,13-\,\,\,\,\,\,1=\,\,\,12$  $\varphi (\,\,\,75,\,\,\,9)=\varphi (\,\,\,75,\,\,\,8)-\varphi (\,\,\,3,8)=\,\,\,=\,\,\,14-\,\,\,\,\,\,1=\,\,\,13$  $\varphi (\,\,\,75,\,\,\,8)=\varphi (\,\,\,75,\,\,\,7)-\varphi (\,\,\,3,7)=\,\,\,=\,\,\,15-\,\,\,\,\,\,1=\,\,\,14$  $\varphi (\,\,\,75,\,\,\,7)=\varphi (\,\,\,75,\,\,\,6)-\varphi (\,\,\,4,6)=\,\,\,=\,\,\,16-\,\,\,\,\,\,1=\,\,\,15$  $\varphi (\,\,\,75,\,\,\,6)=\varphi (\,\,\,75,\,\,\,5)-\varphi (\,\,\,5,5)=\,\,\,=\,\,\,17-\,\,\,\,\,\,1=\,\,\,16$  $\varphi (\,\,\,75,\,\,\,5)=\varphi (\,\,\,75,\,\,\,4)-\varphi (\,\,\,6,4)=\,\,\,=\,\,\,18-\,\,\,\,\,\,1=\,\,\,17$  $\varphi (\,\,\,75,\,\,\,4)=\varphi (\,\,\,75,\,\,\,3)-\varphi (10,3)=\,\,\,=\,\,\,20-\,\,\,\,\,\,2=\,\,\,18$  $\varphi (\,\,\,75,\,\,\,3)=\varphi (\,\,\,75,\,\,\,2)-\varphi (15,2)=\,\,\,=\,\,\,25-\,\,\,\,\,\,5=\,\,\,20$  $\varphi (\,\,\,75,\,\,\,2)=\varphi (\,\,\,75,\,\,\,1)-\varphi (25,1)=\,\,\,=\,\,\,38-\,\,\,13=\,\,\,25$  $\varphi (\,\,\,15,\,\,\,2)=\varphi (15,\,\,\,1)-\varphi (5,1)=\,\,\,\,\,\,8-\,\,\,\,\,\,3$	$=$	$\,\,\,\,\,\,5$

$\varphi (\,\,\,10,3)=\varphi (10,2)-\varphi (2,2)=\,\,\,3-\,\,\,\,\,\,1$  $\varphi (\,\,\,10,2)=\varphi (10,1)-\varphi (3,1)=\,\,\,5-\,\,\,\,\,\,2$	$=$ $=$	$\,\,\,2$ $\,\,\,3$
$\varphi (62,11)=\varphi (62,10)-\varphi (\,\,\,2,10)=\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,8$  $\varphi (62,10)=\varphi (62,\,\,\,9)-\varphi (\,\,\,2,\,\,\,9)=\,\,\,\,\,\,10\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,9$  $\varphi (62,\,\,\,9)=\varphi (62,\,\,\,8)-\varphi (\,\,\,2,\,\,\,8)=\,\,\,\,\,\,11\,\,\,\,\,\,\,-\,\,\,1=\,\,\,10$  $\varphi (62,\,\,\,8)=\varphi (62,\,\,\,7)-\varphi (\,\,\,3,\,\,\,7)=\,\,\,\,\,\,12\,\,\,\,\,\,\,-\,\,\,1=\,\,\,11$  $\varphi (62,\,\,\,7)=\varphi (62,\,\,\,6)-\varphi (\,\,\,3,\,\,\,6)=\,\,\,\,\,\,13\,\,\,\,\,\,\,-\,\,\,1=\,\,\,12$  $\varphi (62,\,\,\,6)=\varphi (62,\,\,\,5)-\varphi (\,\,\,4,\,\,\,5)=\,\,\,\,\,\,14\,\,\,\,\,\,\,-\,\,\,1=\,\,\,13$  $\varphi (62,\,\,\,5)=\varphi (62,\,\,\,4)-\varphi (\,\,\,5,\,\,\,4)=\,\,\,\,\,\,15\,\,\,\,\,\,\,-\,\,\,1=\,\,\,14$  $\varphi (62,\,\,\,4)=\varphi (62,\,\,\,3)-\varphi (\,\,\,8,\,\,\,3)=\,\,\,\,\,\,17\,\,\,\,\,\,\,-\,\,\,2=\,\,\,15$  $\varphi (62,\,\,\,3)=\varphi (62,\,\,\,2)-\varphi (12,\,\,\,2)=\,\,\,\,\,\,21\,\,\,\,\,\,\,-\,\,\,4=\,\,\,17$  $\varphi (62,\,\,\,2)=\varphi (62,\,\,\,1)-\varphi (20,\,\,\,2)=\,\,\,\,\,\,31\,\,\,\,\,\,\,-10=\,\,\,21$  $\varphi (12,2)=\varphi (12,1)-\varphi (4,1)=\,\,\,6-\,\,\,\,\,\,2$	$=$	$4$
$\varphi (\,\,\,8,3)=\varphi (\,\,\,8,2)-\varphi (1,2)=\,\,\,3-\,\,\,\,\,\,1$  $\varphi (\,\,\,8,2)=\varphi (\,\,\,8,1)-\varphi (2,1)=\,\,\,4-\,\,\,\,\,\,1$	$=$ $=$	$2$ $3$
$\varphi (56,12)=\varphi (56,11)-\varphi (\,\,\,1,11)=\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,5$  $\varphi (56,11)=\varphi (56,10)-\varphi (\,\,\,1,10)=\,\,\,\,\,\,\,\,\,7\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,6$  $\varphi (56,10)=\varphi (56,\,\,\,9)-\varphi (\,\,\,1,\,\,\,9)=\,\,\,\,\,\,\,\,\,8\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,7$  $\varphi (56,\,\,\,9)=\varphi (56,\,\,\,8)-\varphi (\,\,\,2,\,\,\,8)=\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,8$  $\varphi (56,\,\,\,8)=\varphi (56,\,\,\,7)-\varphi (\,\,\,2,\,\,\,7)=\,\,\,\,\,10\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,9$  $\varphi (56,\,\,\,7)=\varphi (56,\,\,\,6)-\varphi (\,\,\,3,\,\,\,6)=\,\,\,\,\,11\,\,\,\,\,\,\,-\,\,\,1=\,\,\,10$  $\varphi (56,\,\,\,6)=\varphi (56,\,\,\,5)-\varphi (\,\,\,4,\,\,\,5)=\,\,\,\,\,12\,\,\,\,\,\,\,-\,\,\,1=\,\,\,11$  $\varphi (56,\,\,\,5)=\varphi (56,\,\,\,4)-\varphi (\,\,\,5,\,\,\,4)=\,\,\,\,\,13\,\,\,\,\,\,\,-\,\,\,1=\,\,\,12$  $\varphi (56,\,\,\,4)=\varphi (56,\,\,\,3)-\varphi (\,\,\,8,\,\,\,3)=\,\,\,\,\,15\,\,\,\,\,\,\,-\,\,\,2=\,\,\,13$  $\varphi (56,\,\,\,3)=\varphi (56,\,\,\,2)-\varphi (11,\,\,\,2)=\,\,\,\,\,19\,\,\,\,\,\,\,-\,\,\,4=\,\,\,15$  $\varphi (56,\,\,\,2)=\varphi (56,\,\,\,1)-\varphi (18,\,\,\,1)=\,\,\,\,\,28\,\,\,\,\,\,\,-\,\,\,9=\,\,\,19$

 Obliczenie podług tej metody dla $n=1,000\,000$, wymaga miejsca i czasu prawie 20 razy tyle, co dla $n=100\,000$; gdyż ${\sqrt {n}}=100$, a $\psi (100)=25;\varphi (1,000\,000,\,\,\,25)$. Ilość więc potrzebnych dzieleń i odejmowań bardzo się powiększa.
 Skraca się nieco rachunek powyższy przez uwzględnienie koła luk w numeracyi, jakie powstają przez wyjęcie liczb podzielnych przez liczby pierwsze. Koła te tworzą się podług wzoru

$\varphi (p_{1}.p_{2}..\,\,\,..p_{n},n)=(p_{1}-1)(p_{2}-1)(\,\,\,.\,\,\,.\,\,\,.\,\,\,)(p_{n}-1)$,

[26] czyli $\varphi (2,1)=2-1=\,\,\,1$

 $2.3=\,\,\,6;\,\,\,\varphi (6,\,\,\,2)=(2-1)(3-1)=2$
 $2.3.5=30;\,\,\,\varphi (30,3)=2(5-1)=8;\,\,\,\varphi (60,3)=16;\,\,\,\varphi (90,\,\,\,3)=24;\dots$
 $2.3.5.7=210;\,\,\,\varphi (210,4)=8(7-1)=48;\,\,\,\varphi (420,4)=96;\,\,\,\varphi (630,\,\,\,4)=144;\dots$
 $2.3.5.7.11=2310;\,\,\,\varphi (2310,5)=48(11-1)=480;\,\,\,\varphi (4620,5)=960;\,\,\,\varphi (6930,\,\,\,5)=1440\dots$
 $2.3.5.7.11.13=30030;\,\,\,\varphi (30030,6)=480(13-1)=5760;\,\,\,\varphi (30030,6)=960;\,\,\,\varphi (30030,\,\,\,6)=11520,\dots$
$2.3.5.7.11.13.17=510510;\,\,\,\varphi (510510,7)=5760(17-1)=92160$i t. d.

 Odkrycie tych kół przypisują Drowi Meisselowi. To tylko mię dziwi, że Wertheim, wykładając wyżej przytoczoną formułę Meissela, nic o tych kołach nie wspomina. Dr. Hossfeld, gdym mu je zakomunikował, z początku uznał rzecz za nową, a potem doniósł, że Dr. Meissel je wynalazł i opisał w „Mathematische Annalen“ Bd. II , około roku 1870.
 Większe bez porównania od tych kół ułatwienie obliczeń daje tablica analityczna funkcyi $\varphi (m)$. Przytoczę przykład z tablicy z rachunkiem luk $\varphi (n,\,\,\,6)$.
 Za pomocą kół i tablicy rachunek, wyżej wykonany, robi się tak:
$\varphi (100\,000,\,\,\,4)=\varphi (100\,000,\,\,\,6)-\varphi \left({\frac {100\,000}{p_{7}}},6\right)-\varphi \left({\frac {100\,000}{p_{8}}},7\right)-\ldots -\varphi \left({\frac {100\,000}{p_{14}}},13\right)$

$\varphi (100\,000,\,\,\,6)={\frac {100\,000}{30030}}=3.5{\overset {6}{76}}0+[\varphi (9910,\,\,\,6)=1{\overset {6}{9}}01]=17280+1{\overset {6}{9}}01=19181$

$-\varphi \left({\frac {100\,000}{31}},\,\,\,10\right)=\varphi (3225,10)={\overset {6}{61}}8-({\overset {6}{37}}+{\overset {7}{33}}+{\overset {8}{27}}+{\overset {9}{21}}=118)\,\,\,\,\,\,.\,\,\,$

$-\varphi \left({\frac {100\,000}{17}},\,\,\,6\right)=\varphi (5882,\,\,\,6)={\overset {6}{11}}28\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\,\,\,\,\,\,.\,\,\,\,\,\,.\,\,\,\,\,\,.\,\,\,\,\,\,.\,\,\,\,\,\,.\,\,\,\,\,\,$	$-\,\,\,{\overset {6}{11}}28$
$-\varphi \left({\frac {100\,000}{19}},\,\,\,7\right)=\varphi (5263,\,\,\,7)=1{\overset {6}{0}}09-{\overset {6}{59}}=9{\overset {7}{5}}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\,\,\,\,\,\,.\,\,\,$	$-\,\,\,\,\,\,9{\overset {7}{5}}0$
$-\varphi \left({\frac {100\,000}{23}},\,\,\,8\right)=\varphi (4347,\,\,\,8)={\overset {6}{83}}4-({\overset {6}{49}}+{\overset {7}{43}}-92)={\overset {6}{74}}2\,\,\,\,\,\,.\,\,\,$	$-\,\,\,\,\,\,{\overset {8}{74}}2$
$-\varphi \left({\frac {100\,000}{29}},\,\,\,9\right)=\varphi (3448,\,\,\,9)={\overset {6}{664}}-({\overset {6}{41}}+{\overset {7}{36}}-{\overset {8}{28}}=105)\,\,\,\,\,\,.\,\,\,$	$-\,\,\,\,\,\,{\overset {9}{556}}$
$=\,\,\,{\overset {10}{500}}-{\overset {10}{50}}0$
$-\varphi \left({\frac {100\,000}{37}},\,\,\,11\right)=\varphi (2702,11)={\overset {6}{51}}9-({\overset {6}{32}}+{\overset {7}{28}}+{\overset {8}{23}}+{\overset {9}{16}}+{\overset {10}{14}}=113)\,\,\,\,\,\,.\,\,\,$	$=\,\,\,{\overset {11}{406}}-{\overset {11}{406}}$
$-\varphi \left({\frac {100\,000}{41}},\,\,\,12\right)=\varphi (2439,12)={\overset {6}{4}}68-({\overset {6}{29}}+{\overset {7}{25}}+{\overset {8}{20}}+{\overset {9}{15}}+{\overset {10}{12}}+{\overset {11}{8}}=109)\,\,\,\,\,\,.\,\,\,$	$=\,\,\,{\overset {12}{359}}-{\overset {12}{359}}$
$-\varphi \left({\frac {100\,000}{43}},\,\,\,13\right)=\varphi (2325,12)={\overset {6}{4}}45-({\overset {6}{27}}+{\overset {7}{24}}+{\overset {8}{19}}+{\overset {9}{14}}+{\overset {10}{12}}+{\overset {11}{8}}+{\overset {12}{5}}=109)\,\,\,\,\,\,.\,\,\,$	$=\,\,\,{\overset {13}{336}}-{\overset {13}{336}}$ ${\overline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf {-4977} }}$

 A zatem $\varphi (100\,000,\,\,\,14)={\overset {5}{19}}181-4877={\overset {14}{\mathbf {14204} }};$
 W drugiej połowie formuły Meissela tylko człon $-1$ jest naturalnym, bo w numeracyi liczba $1$ przez funkcyę $\varphi (m)$ pozostaje nietkniętą, a w wyrażeniu $\psi (m)$ nie powinna się znajdować.
 Człon zaś: $\sum \limits _{s=1}^{s=\mu }\psi \left({\frac {n}{p_{\scriptstyle {\text{m+s}}}}}\right)$ od wartości $\varphi (n,m)$ odejmuje za wiele i dla tego staje się potrzebną reetytucya przez człony $+m(\mu +1){\overset {\mu (\mu -1)}{2}}$. Jedyną racyą może być chyba potrzeba odróżnienia liczby liczb pierwszych, znajdujących się w pierwiastku sześciennym, $m=\varphi {\sqrt[{3}]{n}}$, od reszty liczb pierwszych, znajdujących się w pierwiastku kwadratowym, gdyż $m+\mu =\varphi {\sqrt {n}}$.
 Liczby podzielne przez liczby pierwsze, wyrażone przez $\mu$, z taką samą łatwością, jak przez sigma, bierze wartość $\varphi ({\frac {n}{p{\scriptstyle {\text{k}}}}}$ całą; funkcya zaś $\varphi (m)$ bierze tę samą wartosć zmniejszoną $\varphi \left({\frac {n}{p{\scriptstyle {\text{k}}}}},k-1\right)=\psi ({\frac {n}{p{\scriptstyle {\text{k}}}}}-(k-2)$^[1].
 Można się o tem przekonać z następujących obliczeń; a najprzód sigma i $\varphi (n,m+s)$.
[28]

$p_{15}=47;\psi \left({\frac {100\,000}{47}}\right)=\psi (2127)=319;$

$\,\,\,\,\,\varphi \left({\frac {100\,000}{47}},14\right)=\varphi \left(2127),14\right)=319-13=306$

$p_{16}=53;\psi \left({\frac {100\,000}{53}}\right)=\psi (1886)=289;$

$\varphi \left(1886,15\right)=289-14=275$

$p_{17}=59;\psi \left({\frac {100\,000}{59}}\right)=\psi (1694)=264;$

$\varphi \left(1694,16\right)=264-15=249$

$p_{18}=61;\psi \left({\frac {100\,000}{61}}\right)=\psi (1639)=259;$

$\varphi \left(1639,17\right)=259-16=243$

$p_{19}=67;\psi \left({\frac {100\,000}{67}}\right)=\psi (1492)=237;$

$\varphi \left(1492,18\right)=237-17=220$

$p_{20}=71;\psi \left({\frac {100\,000}{71}}\right)=\psi (1408)=222;$

$\varphi \left(1408,19\right)=222-18=204$

$p_{21}=73;\psi \left({\frac {100\,000}{73}}\right)=\psi (1369)=219;$

$\varphi \left(1369,20\right)=219-19=200$

$p_{22}=79;\psi \left({\frac {100\,000}{79}}\right)=\psi (1268)=205;$

$\varphi \left(1268,21\right)=205-20=185$

$p_{23}=83;\psi \left({\frac {100\,000}{83}}\right)=\psi (1204)=197;$

$\varphi \left(1204,22\right)=197-21=176$

$p_{24}=89;\psi \left({\frac {100\,000}{89}}\right)=\psi (1123)=188;$

$\varphi \left(1123,23\right)=188-22=166$

$p_{25}=97;\psi \left({\frac {100\,000}{97}}\right)=\psi (1030)=172;$

$\varphi \left(1030,24\right)=172-23=149$

$p_{26}=101;\psi \left({\frac {100\,000}{101}}\right)=\psi (990)=166;$

$\varphi \left(990,25\right)=166-24=142$

$p_{27}=103;\psi \left({\frac {100\,000}{103}}\right)=\psi (970)=163;$

$\varphi \left(970,26\right)=163-25=138$

$p_{28}=107;\psi \left({\frac {100\,000}{107}}\right)=\psi (934)=158;$

$\varphi \left(934,27\right)=158-26=132$

$p_{29}=109;\psi \left({\frac {100\,000}{109}}\right)=\psi (917)=156;$

$\varphi \left(917,28\right)=156-27=129$

$p_{30}=113;\psi \left({\frac {100\,000}{113}}\right)=\psi (884)=153;$

$\varphi \left(884,29\right)=153-28=125$

$p_{31}=127;\psi \left({\frac {100\,000}{127}}\right)=\psi (787)=138;$

$\varphi \left(787,30\right)=138-29=109$

$p_{32}=131;\psi \left({\frac {100\,000}{131}}\right)=\psi (763)=135;$

$\varphi \left(763,31\right)=135-30=105$

$p_{33}=137;\psi \left({\frac {100\,000}{137}}\right)=\psi (729)=129;$

$\varphi \left(729,32\right)=129-31=\,98$

$p_{34}=139;\psi \left({\frac {100\,000}{139}}\right)=\psi (719)=128;$

$\varphi \left(719,33\right)=128-32=\,96$

$p_{35}=149;\psi \left({\frac {100\,000}{149}}\right)=\psi (671)=121;$

$\varphi \left(671,34\right)=121-33=\,88$

$p_{36}=151;\psi \left({\frac {100\,000}{151}}\right)=\psi (662)=121;$

$\varphi \left(662,35\right)=121-34=\,87$

$p_{37}=157;\psi \left({\frac {100\,000}{157}}\right)=\psi (636)=115;$

$\varphi \left(636,36\right)=115-35=\,80$

$p_{38}=163;\psi \left({\frac {100\,000}{163}}\right)=\psi (613)=112;$

$\varphi \left(613,37\right)=112-36=\,76$

$p_{39}=167;\psi \left({\frac {100\,000}{167}}\right)=\psi (598)=108;$

$\varphi \left(598,38\right)=108-37=\,71$

$p_{40}=173;\psi \left({\frac {100\,000}{173}}\right)=\psi (578)=106;$

$\varphi \left(578,39\right)=106-38=\,68$

$p_{41}=179;\psi \left({\frac {100\,000}{179}}\right)=\psi (558)=102;$

$\varphi \left(558,50\right)=102-39=\,63$

$p_{42}=181;\psi \left({\frac {100\,000}{181}}\right)=\psi (552)=101;$

$\varphi \left(552,51\right)=101-40=\,61$

$p_{43}=191;\psi \left({\frac {100\,000}{191}}\right)=\psi (523)=\,99;$

$\varphi \left(523,52\right)=99\,-41=\,58$

$p_{44}=193;\psi \left({\frac {100\,000}{193}}\right)=\psi (518)=\,97;$

$\varphi \left(518,53\right)=97\,-42=\,55$

$p_{45}=197;\psi \left({\frac {100\,000}{197}}\right)=\psi (507)=\,96;$

$\varphi \left(507,54\right)=96\,-43=\,53$

$p_{46}=199;\psi \left({\frac {100\,000}{199}}\right)=\psi (502)=\,95;$

$\varphi \left(502,55\right)=95\,-44=\,51$

$p_{47}=211;\psi \left({\frac {100\,000}{211}}\right)=\psi (473)=\,91;$

$\varphi \left(473,46\right)=91\,-45=\,46$

$p_{48}=223;\psi \left({\frac {100\,000}{223}}\right)=\psi (448)=\,86;$

$\varphi \left(448,47\right)=86\,-46=\,40$

$p_{49}=227;\psi \left({\frac {100\,000}{227}}\right)=\psi (440)=\,85;$

$\varphi \left(440,48\right)=85\,-47=\,38$

$p_{50}=229;\psi \left({\frac {100\,000}{229}}\right)=\psi (436)=\,84;$

$\varphi \left(436,49\right)=84\,-48=\,36$

$p_{51}=233;\psi \left({\frac {100\,000}{233}}\right)=\psi (429)=\,82;$

$\varphi \left(429,50\right)=82\,-49=\,33$

$p_{51}=233;\psi \left({\frac {100\,000}{233}}\right)=\psi (418)=\,80;$

$\varphi \left(418,51\right)=80\,-50=\,30$

$p_{52}=239;\psi \left({\frac {100\,000}{239}}\right)=\psi (418)=\,80;$

$\varphi \left(418,51\right)=80\,-50=\,30$

$p_{53}=241;\psi \left({\frac {100\,000}{241}}\right)=\psi (414)=\,80;$

$\varphi \left(414,52\right)=80\,-51=\,29$

$p_{54}=251;\psi \left({\frac {100\,000}{251}}\right)=\psi (398)=\,78;$

$\varphi \left(398,53\right)=78\,-52=\,26$

$p_{55}=257;\psi \left({\frac {100\,000}{257}}\right)=\psi (389)=\,77;$

$\varphi \left(389,54\right)=77\,-53=\,24$

$p_{56}=263;\psi \left({\frac {100\,000}{263}}\right)=\psi (380)=\,75;$

$\varphi \left(380,55\right)=75\,-54=\,21$

$p_{57}=269;\psi \left({\frac {100\,000}{269}}\right)=\psi (371)=\,73;$

$\varphi \left(371,56\right)=73\,-55=\,18$

$p_{58}=271;\psi \left({\frac {100\,000}{271}}\right)=\psi (369)=\,73;$

$\varphi \left(369,57\right)=73\,-56=\,17$

$p_{59}=277;\psi \left({\frac {100\,000}{277}}\right)=\psi (361)=\,72;$

$\varphi \left(361,58\right)=72\,-57=\,15$

$p_{60}=281;\psi \left({\frac {100\,000}{281}}\right)=\psi (355)=\,71;$

$\varphi \left(355,59\right)=71\,-58=\,13$

$p_{61}=283;\psi \left({\frac {100\,000}{283}}\right)=\psi (353)=\,71;$

$\varphi \left(353,60\right)=71\,-59=\,12$

$p_{62}=293;\psi \left({\frac {100\,000}{293}}\right)=\psi (341)=68;$

$\psi \left(341,61\right)=68\,-60=\,\,8$

$p_{63}=307;\psi \left({\frac {100\,000}{307}}\right)=\psi (325)=66;$

$\psi \left(325,62\right)=66\,-61=\,\,5$

$p_{64}=311;\psi \left({\frac {100\,000}{311}}\right)=\psi (321)=66;$

$\psi \left(321,63\right)=66\,-62=\,\,4$

$p_{65}=313;\psi \left({\frac {100\,000}{313}}\right)=\psi (319)=66;$
 ${\overline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf {6614} }}$

$\psi \left(319,64\right)=66\,-63=\,\,3$
${\overline {\,\,\,\,\,\,\,\,\mathbf {4676} \,\,}}$

 A zatem — $\sum \limits _{s=1}^{s=51}\psi \left({\frac {100\,000}{p_{14+s}}}\right)=-\mathbf {\,6614}$

 $+m\left(\mu +1\right)=14.\,52=\mathbf {728}$

 $+{\frac {\mu (\mu -1)}{2}}={\frac {51.\,50}{2}}={\frac {2550}{2}}=\mathbf {1275}$

$\psi (100\,000)=\varphi (100\,000,14)+14.\,52+{\frac {51.\,50}{2}}-1-\sum \limits _{s=1}^{s=51}\psi \left({\frac {100\,000}{p_{14+s}}}\right)=14204+728+1275-1-6614=16102-6615=\mathbf {9592}$

 Funkcya zaś $\varphi (m)$ bez sigmy i bez jej restytucyi daje liczbę liczb pierwszych, do której należy dodać usunięte przez nią [funkcyą $\varphi (m)$] liczb pierwszych 65, a odjąć 1; czyli $\psi (100\,000)=\varphi (100\,000,14)-\varphi (100\,000,65)+65-1=14204-4676-1+65=14269-4677=\mathbf {9592} .$
 Stąd wnioskuję, że wzór $\psi (n)=\varphi (n,\psi {\sqrt {n}})+\psi {\sqrt {n}}-1$ jest prostszy i naturalniejszy od Meisselowskiego i nie trudniejszy do obliczenia.

1. Wyrażenie $p_{k}=p_{m+s}$.

Tekst jest własnością publiczną (public domain). Szczegóły licencji na stronie autora: Antanas Baranauskas.