A244963 a(n) = sigma(n) - n * Product_{p|n, p prime} (1 + 1/p).

0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 4, 0, 0, 0, 7, 0, 3, 0, 6, 0, 0, 0, 12, 1, 0, 4, 8, 0, 0, 0, 15, 0, 0, 0, 19, 0, 0, 0, 18, 0, 0, 0, 12, 6, 0, 0, 28, 1, 3, 0, 14, 0, 12, 0, 24, 0, 0, 0, 24, 0, 0, 8, 31, 0, 0, 0, 18, 0, 0, 0, 51, 0, 0, 4, 20, 0, 0, 0, 42, 13

Offset: 1

Author

Omar E. Pol, Jul 15 2014

Keywords

• nonn
• easy

Comments

a(n) = 0 if and only if n is a squarefree number (A005117), otherwise a(n) > 0.

If n is semiprime, then a(n) = 1+floor(sqrt(n))-ceiling(sqrt(n)). - Wesley Ivan Hurt, Dec 25 2016

Links

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

Crossrefs

Cf. A000203 (sigma), A001615 (Dedekind psi), A005117 (positions of zeros), A013929, A049417.

Programs

Maple

A244963:= n -> numtheory:-sigma(n) - n*mul(1+1/t[1],t=ifactors(n)[2]):
seq(A244963(n),n=1..1000); # Robert Israel, Jul 15 2014

Mathematica

nn = 200;Table[Sum[d, {d, Divisors[n]}], {n, 1, nn}] -
Table[Sum[n/d Abs[MoebiusMu[d]], {d, Divisors[n]}], {n, 1, nn}] (* Geoffrey Critzer, Mar 18 2015 *)

PARI

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
A244963(n) = (sigma(n) - A001615(n)); \\ Antti Karttunen, Nov 22 2017

Formula

a(n) = A000203(n) - A001615(n).

Sum_{k=1..n} a(k) ~ c*n^2 + O(n*log(n)), where c = Pi^2/12 - 15/(2*Pi^2) = 0.062558... - Amiram Eldar, Mar 02 2021