

A069088


a(n) = Sum_{dn} 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.


3



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, 8, 32, 30, 72, 32, 9, 48, 54, 48, 20, 38, 60, 56, 36, 42, 96, 44, 48, 30, 72, 48, 28, 9, 21, 72, 56, 54, 24, 72, 48, 80, 90, 60, 96, 62, 96, 40, 10, 84, 144, 68, 72, 96, 144, 72, 30, 74
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OFFSET

1,2


COMMENTS

Multiplicative because it is the inverse Moebius transform of A007913 which is multiplicative.  Christian G. Bower, May 17 2005.


LINKS

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


FORMULA

G.f.: Sum_{k>=1} core(k)*x^k/(1x^k).  Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(2*s)*zeta(s)*zeta(s1)/zeta(2*s2).  R. J. Mathar, Oct 31 2011
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 180.  Vaclav Kotesovec, Feb 01 2019
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


MATHEMATICA

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 *)


PROG

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


CROSSREFS

Cf. A007913.
Sequence in context: A265887 A008473 A326043 * A178450 A325973 A019462
Adjacent sequences: A069085 A069086 A069087 * A069089 A069090 A069091


KEYWORD

easy,nonn,mult


AUTHOR

Benoit Cloitre, Apr 05 2002


STATUS

approved



