φ ( 10 , 3 ) = φ ( 10 , 2 ) − φ ( 2 , 2 ) = 3 − 1 {\displaystyle \varphi (\,\,\,10,3)=\varphi (10,2)-\varphi (2,2)=\,\,\,3-\,\,\,\,\,\,1} φ ( 10 , 2 ) = φ ( 10 , 1 ) − φ ( 3 , 1 ) = 5 − 2 {\displaystyle \varphi (\,\,\,10,2)=\varphi (10,1)-\varphi (3,1)=\,\,\,5-\,\,\,\,\,\,2}
= {\displaystyle =} = {\displaystyle =}
2 {\displaystyle \,\,\,2} 3 {\displaystyle \,\,\,3}
φ ( 62 , 11 ) = φ ( 62 , 10 ) − φ ( 2 , 10 ) = 9 − 1 = 8 {\displaystyle \varphi (62,11)=\varphi (62,10)-\varphi (\,\,\,2,10)=\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,8} φ ( 62 , 10 ) = φ ( 62 , 9 ) − φ ( 2 , 9 ) = 10 − 1 = 9 {\displaystyle \varphi (62,10)=\varphi (62,\,\,\,9)-\varphi (\,\,\,2,\,\,\,9)=\,\,\,\,\,\,10\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,9} φ ( 62 , 9 ) = φ ( 62 , 8 ) − φ ( 2 , 8 ) = 11 − 1 = 10 {\displaystyle \varphi (62,\,\,\,9)=\varphi (62,\,\,\,8)-\varphi (\,\,\,2,\,\,\,8)=\,\,\,\,\,\,11\,\,\,\,\,\,\,-\,\,\,1=\,\,\,10} φ ( 62 , 8 ) = φ ( 62 , 7 ) − φ ( 3 , 7 ) = 12 − 1 = 11 {\displaystyle \varphi (62,\,\,\,8)=\varphi (62,\,\,\,7)-\varphi (\,\,\,3,\,\,\,7)=\,\,\,\,\,\,12\,\,\,\,\,\,\,-\,\,\,1=\,\,\,11} φ ( 62 , 7 ) = φ ( 62 , 6 ) − φ ( 3 , 6 ) = 13 − 1 = 12 {\displaystyle \varphi (62,\,\,\,7)=\varphi (62,\,\,\,6)-\varphi (\,\,\,3,\,\,\,6)=\,\,\,\,\,\,13\,\,\,\,\,\,\,-\,\,\,1=\,\,\,12} φ ( 62 , 6 ) = φ ( 62 , 5 ) − φ ( 4 , 5 ) = 14 − 1 = 13 {\displaystyle \varphi (62,\,\,\,6)=\varphi (62,\,\,\,5)-\varphi (\,\,\,4,\,\,\,5)=\,\,\,\,\,\,14\,\,\,\,\,\,\,-\,\,\,1=\,\,\,13} φ ( 62 , 5 ) = φ ( 62 , 4 ) − φ ( 5 , 4 ) = 15 − 1 = 14 {\displaystyle \varphi (62,\,\,\,5)=\varphi (62,\,\,\,4)-\varphi (\,\,\,5,\,\,\,4)=\,\,\,\,\,\,15\,\,\,\,\,\,\,-\,\,\,1=\,\,\,14} φ ( 62 , 4 ) = φ ( 62 , 3 ) − φ ( 8 , 3 ) = 17 − 2 = 15 {\displaystyle \varphi (62,\,\,\,4)=\varphi (62,\,\,\,3)-\varphi (\,\,\,8,\,\,\,3)=\,\,\,\,\,\,17\,\,\,\,\,\,\,-\,\,\,2=\,\,\,15} φ ( 62 , 3 ) = φ ( 62 , 2 ) − φ ( 12 , 2 ) = 21 − 4 = 17 {\displaystyle \varphi (62,\,\,\,3)=\varphi (62,\,\,\,2)-\varphi (12,\,\,\,2)=\,\,\,\,\,\,21\,\,\,\,\,\,\,-\,\,\,4=\,\,\,17} φ ( 62 , 2 ) = φ ( 62 , 1 ) − φ ( 20 , 2 ) = 31 − 10 = 21 {\displaystyle \varphi (62,\,\,\,2)=\varphi (62,\,\,\,1)-\varphi (20,\,\,\,2)=\,\,\,\,\,\,31\,\,\,\,\,\,\,-10=\,\,\,21} φ ( 12 , 2 ) = φ ( 12 , 1 ) − φ ( 4 , 1 ) = 6 − 2 {\displaystyle \varphi (12,2)=\varphi (12,1)-\varphi (4,1)=\,\,\,6-\,\,\,\,\,\,2}
= {\displaystyle =}
4 {\displaystyle 4}
φ ( 8 , 3 ) = φ ( 8 , 2 ) − φ ( 1 , 2 ) = 3 − 1 {\displaystyle \varphi (\,\,\,8,3)=\varphi (\,\,\,8,2)-\varphi (1,2)=\,\,\,3-\,\,\,\,\,\,1} φ ( 8 , 2 ) = φ ( 8 , 1 ) − φ ( 2 , 1 ) = 4 − 1 {\displaystyle \varphi (\,\,\,8,2)=\varphi (\,\,\,8,1)-\varphi (2,1)=\,\,\,4-\,\,\,\,\,\,1}
2 {\displaystyle 2} 3 {\displaystyle 3}
φ ( 56 , 12 ) = φ ( 56 , 11 ) − φ ( 1 , 11 ) = 6 − 1 = 5 {\displaystyle \varphi (56,12)=\varphi (56,11)-\varphi (\,\,\,1,11)=\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,5} φ ( 56 , 11 ) = φ ( 56 , 10 ) − φ ( 1 , 10 ) = 7 − 1 = 6 {\displaystyle \varphi (56,11)=\varphi (56,10)-\varphi (\,\,\,1,10)=\,\,\,\,\,\,\,\,\,7\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,6} φ ( 56 , 10 ) = φ ( 56 , 9 ) − φ ( 1 , 9 ) = 8 − 1 = 7 {\displaystyle \varphi (56,10)=\varphi (56,\,\,\,9)-\varphi (\,\,\,1,\,\,\,9)=\,\,\,\,\,\,\,\,\,8\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,7} φ ( 56 , 9 ) = φ ( 56 , 8 ) − φ ( 2 , 8 ) = 9 − 1 = 8 {\displaystyle \varphi (56,\,\,\,9)=\varphi (56,\,\,\,8)-\varphi (\,\,\,2,\,\,\,8)=\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,8} φ ( 56 , 8 ) = φ ( 56 , 7 ) − φ ( 2 , 7 ) = 10 − 1 = 9 {\displaystyle \varphi (56,\,\,\,8)=\varphi (56,\,\,\,7)-\varphi (\,\,\,2,\,\,\,7)=\,\,\,\,\,10\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,9} φ ( 56 , 7 ) = φ ( 56 , 6 ) − φ ( 3 , 6 ) = 11 − 1 = 10 {\displaystyle \varphi (56,\,\,\,7)=\varphi (56,\,\,\,6)-\varphi (\,\,\,3,\,\,\,6)=\,\,\,\,\,11\,\,\,\,\,\,\,-\,\,\,1=\,\,\,10} φ ( 56 , 6 ) = φ ( 56 , 5 ) − φ ( 4 , 5 ) = 12 − 1 = 11 {\displaystyle \varphi (56,\,\,\,6)=\varphi (56,\,\,\,5)-\varphi (\,\,\,4,\,\,\,5)=\,\,\,\,\,12\,\,\,\,\,\,\,-\,\,\,1=\,\,\,11} φ ( 56 , 5 ) = φ ( 56 , 4 ) − φ ( 5 , 4 ) = 13 − 1 = 12 {\displaystyle \varphi (56,\,\,\,5)=\varphi (56,\,\,\,4)-\varphi (\,\,\,5,\,\,\,4)=\,\,\,\,\,13\,\,\,\,\,\,\,-\,\,\,1=\,\,\,12} φ ( 56 , 4 ) = φ ( 56 , 3 ) − φ ( 8 , 3 ) = 15 − 2 = 13 {\displaystyle \varphi (56,\,\,\,4)=\varphi (56,\,\,\,3)-\varphi (\,\,\,8,\,\,\,3)=\,\,\,\,\,15\,\,\,\,\,\,\,-\,\,\,2=\,\,\,13} φ ( 56 , 3 ) = φ ( 56 , 2 ) − φ ( 11 , 2 ) = 19 − 4 = 15 {\displaystyle \varphi (56,\,\,\,3)=\varphi (56,\,\,\,2)-\varphi (11,\,\,\,2)=\,\,\,\,\,19\,\,\,\,\,\,\,-\,\,\,4=\,\,\,15} φ ( 56 , 2 ) = φ ( 56 , 1 ) − φ ( 18 , 1 ) = 28 − 9 = 19 {\displaystyle \varphi (56,\,\,\,2)=\varphi (56,\,\,\,1)-\varphi (18,\,\,\,1)=\,\,\,\,\,28\,\,\,\,\,\,\,-\,\,\,9=\,\,\,19}
Obliczenie podług tej metody dla n = 1 , 000 000 {\displaystyle n=1,000\,000} , wymaga miejsca i czasu prawie 20 razy tyle, co dla n = 100 000 {\displaystyle n=100\,000} ; gdyż n = 100 {\displaystyle {\sqrt {n}}=100} , a ψ ( 100 ) = 25 ; φ ( 1 , 000 000 , 25 ) {\displaystyle \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