A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

0, 0, 0, 3, 0, 0, 0, 9, 6, 0, 0, 12, 0, 0, 0, 21, 0, 18, 0, 18, 0, 0, 0, 36, 15, 0, 24, 24, 0, 0, 0, 45, 0, 0, 0, 60, 0, 0, 0, 54, 0, 0, 0, 36, 36, 0, 0, 84, 28, 45, 0, 42, 0, 72, 0, 72, 0, 0, 0, 72, 0, 0, 48, 93, 0, 0, 0, 54, 0, 0, 0, 144, 0, 0, 60, 60, 0, 0, 0, 126, 78, 0, 0, 96, 0, 0, 0, 108, 0, 108, 0, 72, 0, 0, 0, 180, 0, 84, 72

Offset: 1

Author

Antti Karttunen, Jun 02 2019

Keywords

• nonn

Examples

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, non-unitary divisors are 2, 3, 6, 12 and 18 so A048146(36) = 2+3+6+12+18 = 41, while the nonsquarefree divisors are 4, 9, 12, 18 and 36, so A162296(36) = 4+9+12+18+36 = 79, thus a(36) = (41+79)/2 = 60.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000203, A005117, A034448, A048146, A162296, A325973, A325975, A325978.

Cf. A002117, A013661.

Programs

PARI

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325974(n) = ((A048146(n)+A162296(n))/2);

Formula

a(n) = (1/2)*(A048146(n) + A162296(n)).

a(n) = A000203(n) - A325973(n).

a(n) = n - A325978(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)*(1/2 - 1/(4*zeta(3))) - 1/4 = 0.2303588390... . - Amiram Eldar, Feb 22 2024