A065764 Sum of divisors of square numbers.

1, 7, 13, 31, 31, 91, 57, 127, 121, 217, 133, 403, 183, 399, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 1281, 1093, 1767, 871, 2821, 993, 2047, 1729, 2149, 1767, 3751, 1407, 2667, 2379, 3937, 1723, 5187, 1893, 4123, 3751, 3871, 2257, 6643

Offset: 1

Author

Labos Elemer, Nov 19 2001

Keywords

• nonn
• easy
• mult

Comments

Unlike A065765, the sums of divisors of squares give remainders r=1,3,5 modulo 6: sigma(4)==1, sigma(49)==3, sigma(2401)==5 (mod 6). See also A097022.

a(n) is also the number of ordered pairs of positive integers whose LCM is n, (see LeVeque). - Enrique Pérez Herrero, Aug 26 2013

Main diagonal of A319526. - Omar E. Pol, Sep 25 2018

Subsequence of primes is A023195 \ {3}; also, 31 is the only known prime to be twice in the data because 31 = sigma(16) = sigma(25) (see A119598 and Goormaghtigh conjecture link). - Bernard Schott, Jan 17 2021

References

• W. J. LeVeque, Fundamentals of Number Theory, pp. 125 Problem 4, Dover NY 1996.

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• Wikipedia, Goormaghtigh conjecture.

Crossrefs

Cf. A000203, A000290, A028982, A319526.

Programs

GAP

a:=List([1..50],n->Sigma(n^2));; Print(a); # Muniru A Asiru, Jan 01 2019

Magma

[SumOfDivisors(n^2): n in [1..48]]; // Bruno Berselli, Apr 12 2011

Maple

with(numtheory): [sigma(n^2)$n=1..50]; # Muniru A Asiru, Jan 01 2019

Mathematica

Table[Plus@@Divisors[n^2], {n, 48}] (* Alonso del Arte, Feb 24 2012 *)
f[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

MuPAD

numlib::sigma(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI

a(n) = sigma(n^2); \\ Harry J. Smith, Oct 30 2009

Python

from math import prod
from sympy import factorint
def A065764(n): return prod((p**((e<<1)+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

Sage

[sigma(n^2,1)for n in range(1,49)] # Zerinvary Lajos, Jun 13 2009

Formula

a(n) = sigma(n^2) = A000203(A000290(n)).

Multiplicative with a(p^e) = (p^(2*e+1)-1)/(p-1). - Vladeta Jovovic, Dec 01 2001

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/zeta(2*s-2), inverse Mobius transform of A000082. - R. J. Mathar, Mar 06 2011

Dirichlet convolution of A001157 by the absolute terms of A055615. Also the Dirichlet convolution of A048250 by A000290. - R. J. Mathar, Apr 12 2011

a(n) = Sum_{d|n} d*Psi(d), where Psi is A001615. - Enrique Pérez Herrero, Feb 25 2012

a(n) >= (n+1) * sigma(n) - n, where sigma is A000203, equality holds if n is in A000961. - Enrique Pérez Herrero, Apr 21 2012

Sum_{k=1..n} a(k) ~ 5*Zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, Jan 30 2019

Sum_{k>=1} 1/a(k) = 1.3947708738535614499846243600124612760835313454790187655653356563282177118... - Vaclav Kotesovec, Sep 20 2020