A318305 a(n) = Product_{primes p dividing n} p - Product_{primes p dividing n} (p-1).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Aug 26 2018

Keywords

• nonn

Examples

			For n = 45 = 3^2 * 5, the prime factors are 3 and 5, thus a(45) = (3*5) - (2*4) = 15 - 8 = 7.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065463 - A307868 = 0.232761... . - _Amiram Eldar_, Dec 07 2023
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A003557, A007947, A051953, A083254, A173557, A318304.

Cf. A065463, A307868.

Programs

PARI

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A051953(n) = (n - eulerphi(n));
A318305(n) = A051953(n)/A003557(n);

PARI

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A318305(n) = (A007947(n) - A173557(n));

Formula

a(n) = A051953(n)/A003557(n) = A007947(n) - A173557(n) = A173557(n) - A318304(n).

Extensions

Corrected the notation in the definition - Antti Karttunen, Feb 03 2024