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%I #36 Mar 02 2021 06:00:50
%S 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,
%T 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,
%U 31,0,0,0,18,0,0,0,51,0,0,4,20,0,0,0,42,13
%N a(n) = sigma(n) - n * Product_{p|n, p prime} (1 + 1/p).
%C a(n) = 0 if and only if n is a squarefree number (A005117), otherwise a(n) > 0.
%C If n is semiprime, then a(n) = 1+floor(sqrt(n))-ceiling(sqrt(n)). - _Wesley Ivan Hurt_, Dec 25 2016
%H Antti Karttunen, <a href="/A244963/b244963.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = A000203(n) - A001615(n).
%F 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
%p A244963:= n -> numtheory:-sigma(n) - n*mul(1+1/t[1],t=ifactors(n)[2]):
%p seq(A244963(n),n=1..1000); # _Robert Israel_, Jul 15 2014
%t nn = 200;Table[Sum[d, {d, Divisors[n]}], {n, 1, nn}] -
%t Table[Sum[n/d Abs[MoebiusMu[d]], {d, Divisors[n]}], {n, 1, nn}] (* _Geoffrey Critzer_, Mar 18 2015 *)
%o (PARI)
%o A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ This function from _Charles R Greathouse IV_, Sep 09 2014
%o A244963(n) = (sigma(n) - A001615(n)); \\ _Antti Karttunen_, Nov 22 2017
%Y Cf. A000203 (sigma), A001615 (Dedekind psi), A005117 (positions of zeros), A013929, A049417.
%K nonn,easy
%O 1,8
%A _Omar E. Pol_, Jul 15 2014
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