A065827 Sum of squares of divisors of square numbers.

1, 21, 91, 341, 651, 1911, 2451, 5461, 7381, 13671, 14763, 31031, 28731, 51471, 59241, 87381, 83811, 155001, 130683, 221991, 223041, 310023, 280371, 496951, 406901, 603351, 597871, 835791, 708123, 1244061, 924483, 1398101, 1343433, 1760031, 1595601, 2516921

Offset: 1

Author

Vladeta Jovovic, Dec 06 2001

Keywords

• mult
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Harry J. Smith)

Crossrefs

Cf. A001157, A065764.

Programs

Maple

A065827 := proc(n) numtheory[sigma][2](n^2) ; end proc:
seq(A065827(n),n=1..20) ; # R. J. Mathar, Apr 01 2011

Mathematica

DivisorSigma[2,#]&/@(Range[40]^2) (* Harvey P. Dale, May 18 2011 *)
f[p_, e_] := (p^(4*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

PARI

{ for (n=1, 500, a=sigma(n^2, 2); write("b065827.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 01 2009

Sage

[sigma(n^2,2)for n in range(1,34)] # Zerinvary Lajos, Jun 13 2009

Formula

Multiplicative with a(p^e) = (p^(4*e+2)-1)/(p^2-1).

a(n) = A001157(n^2). - R. J. Mathar, Mar 31 2011

Dirichlet g.f. zeta(s)*zeta(s-2)*zeta(s-4)/zeta(2s-4). Dirichlet convolution of A001159 by the arithmetic function with terms n^2*A008966(n). - R. J. Mathar, Mar 31 2011

Sum_{k=1..n} a(k) ~ 189 * Zeta(3) * Zeta(5) * n^5 / Pi^6. - Vaclav Kotesovec, Feb 01 2019

Sum_{k>=1} 1/a(k) = 1.06464520174524878494847955427968776606386158167258511428260450690334042955... - Vaclav Kotesovec, Sep 20 2020