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A065764 Sum of divisors of square numbers. 32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

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

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 *)

PROG

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

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

(PARI) { for (n=1, 10000, write("b065764.txt", n, " ", sigma(n^2)) ) } \\ Harry J. Smith, Oct 30 2009

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

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

CROSSREFS

Cf. A000203, A000290, A028982, A319526.

Sequence in context: A133325 A283709 A063583 * A273757 A073473 A272407

Adjacent sequences:  A065761 A065762 A065763 * A065765 A065766 A065767

KEYWORD

nonn,easy,mult

AUTHOR

Labos Elemer, Nov 19 2001

STATUS

approved

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Last modified May 20 21:27 EDT 2019. Contains 323426 sequences. (Running on oeis4.)