A033196 a(n) = n^3 * Product_{p|n, p prime} (1 + 1/p).

1, 12, 36, 96, 150, 432, 392, 768, 972, 1800, 1452, 3456, 2366, 4704, 5400, 6144, 5202, 11664, 7220, 14400, 14112, 17424, 12696, 27648, 18750, 28392, 26244, 37632, 25230, 64800, 30752, 49152, 52272, 62424, 58800, 93312, 52022, 86640

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A000082, A001615.

Programs

Mathematica

a[n_] := n*DivisorSum[n, MoebiusMu[n/#] DivisorSigma[1, #^2]&]; Array[a, 40] (* Jean-François Alcover, Dec 02 2015 *)

PARI

a(n)=direuler(p=2,n,(1+p^2*X)/(1-p^3*X))[n]

PARI

a(n)=sumdiv(n,d,moebius(d)*sigma(n^3/d^2)) \\ Benoit Cloitre, Feb 16 2008

Formula

Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(2*s-4).

a(n) = n^2 * A001615(n) = n * A000082(n).

Multiplicative with a(p^e) = p^e*p^(2*e-1)*(p+1). - Vladeta Jovovic, Nov 16 2001

a(n) = Sum_{d|n} mu(d)*sigma(n^3/d^2). - Benoit Cloitre, Feb 16 2008

a(n) = A001615(n^3) = A001615(n^k)/n^(k-3), with k>2. - Enrique Pérez Herrero, Mar 06 2012

Sum_{k=1..n} a(k) ~ 15*n^4 / (4*Pi^2). - Vaclav Kotesovec, Feb 01 2019

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p/((p+1)*(p^3-1))) = 1.1392293101137663761606045655621290749920977339371831842000361508083066155... - Vaclav Kotesovec, Sep 20 2020

Extensions

Additional comments from Michael Somos, May 19 2000