|
%I #21 Feb 03 2024 10:12:49
%S 0,1,1,1,1,4,1,1,1,6,1,4,1,8,7,1,1,4,1,6,9,12,1,4,1,14,1,8,1,22,1,1,
%T 13,18,11,4,1,20,15,6,1,30,1,12,7,24,1,4,1,6,19,14,1,4,15,8,21,30,1,
%U 22,1,32,9,1,17,46,1,18,25,46,1,4,1,38,7,20,17,54,1,6,1,42,1,30,21,44,31,12,1,22,19,24,33,48,23,4,1,8
%N a(n) = Product_{primes p dividing n} p - Product_{primes p dividing n} (p-1).
%H Antti Karttunen, <a href="/A318305/b318305.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = A051953(n)/A003557(n) = A007947(n) - A173557(n) = A173557(n) - A318304(n).
%e For n = 45 = 3^2 * 5, the prime factors are 3 and 5, thus a(45) = (3*5) - (2*4) = 15 - 8 = 7.
%e Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065463 - A307868 = 0.232761... . - _Amiram Eldar_, Dec 07 2023
%o (PARI)
%o A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
%o A051953(n) = (n - eulerphi(n));
%o A318305(n) = A051953(n)/A003557(n);
%o (PARI)
%o A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
%o A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
%o A318305(n) = (A007947(n) - A173557(n));
%Y Cf. A003557, A007947, A051953, A083254, A173557, A318304.
%Y Cf. A065463, A307868.
%K nonn
%O 1,6
%A _Antti Karttunen_, Aug 26 2018
%E Corrected the notation in the definition - _Antti Karttunen_, Feb 03 2024
|
