A069088 - OEIS

%I #26 Sep 03 2020 12:12:40

%S 1,3,4,4,6,12,8,6,5,18,12,16,14,24,24,7,18,15,20,24,32,36,24,24,7,42,

%T 8,32,30,72,32,9,48,54,48,20,38,60,56,36,42,96,44,48,30,72,48,28,9,21,

%U 72,56,54,24,72,48,80,90,60,96,62,96,40,10,84,144,68,72,96,144,72,30,74

%N a(n) = Sum_{d|n} core(d) where d are the divisors of n and where core(d) is the squarefree part of d: the smallest number such that d*core(d) is a square.

%C Multiplicative because it is the inverse Moebius transform of A007913 which is multiplicative. - _Christian G. Bower_, May 17 2005.

%H Antti Karttunen, <a href="/A069088/b069088.txt">Table of n, a(n) for n = 1..16384</a>

%F G.f.: Sum_{k>=1} core(k)*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003

%F Dirichlet g.f.: zeta(2*s)*zeta(s)*zeta(s-1)/zeta(2*s-2). - _R. J. Mathar_, Oct 31 2011

%F Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 180. - _Vaclav Kotesovec_, Feb 01 2019

%F Multiplicative with a(p^e) = (p+1)*(e+1)/2 if e odd, and (p+1)*e/2 + 1 if e even. - _Amiram Eldar_, Sep 03 2020

%t f[p_, e_] := If[OddQ[e], (p + 1)*(e + 1)/2, (p + 1)*e/2 + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 03 2020 *)

%o (PARI) a(n) = sumdiv(n,d, core(d) );

%Y Cf. A007913.

%K easy,nonn,mult

%O 1,2

%A _Benoit Cloitre_, Apr 05 2002