|
%I #84 Oct 25 2023 14:25:46
%S 1,7,13,31,31,91,57,127,121,217,133,403,183,399,403,511,307,847,381,
%T 961,741,931,553,1651,781,1281,1093,1767,871,2821,993,2047,1729,2149,
%U 1767,3751,1407,2667,2379,3937,1723,5187,1893,4123,3751,3871,2257,6643
%N Sum of divisors of square numbers.
%C 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.
%C a(n) is also the number of ordered pairs of positive integers whose LCM is n, (see LeVeque). - _Enrique Pérez Herrero_, Aug 26 2013
%C Main diagonal of A319526. - _Omar E. Pol_, Sep 25 2018
%C 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
%D W. J. LeVeque, Fundamentals of Number Theory, pp. 125 Problem 4, Dover NY 1996.
%H T. D. Noe, <a href="/A065764/b065764.txt">Table of n, a(n) for n=1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>.
%F a(n) = sigma(n^2) = A000203(A000290(n)).
%F Multiplicative with a(p^e) = (p^(2*e+1)-1)/(p-1). - _Vladeta Jovovic_, Dec 01 2001
%F 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
%F Dirichlet convolution of A001157 by the absolute terms of A055615. Also the Dirichlet convolution of A048250 by A000290. - _R. J. Mathar_, Apr 12 2011
%F a(n) = Sum_{d|n} d*Psi(d), where Psi is A001615. - _Enrique Pérez Herrero_, Feb 25 2012
%F a(n) >= (n+1) * sigma(n) - n, where sigma is A000203, equality holds if n is in A000961. - _Enrique Pérez Herrero_, Apr 21 2012
%F Sum_{k=1..n} a(k) ~ 5*Zeta(3) * n^3 / Pi^2. - _Vaclav Kotesovec_, Jan 30 2019
%F Sum_{k>=1} 1/a(k) = 1.3947708738535614499846243600124612760835313454790187655653356563282177118... - _Vaclav Kotesovec_, Sep 20 2020
%p with(numtheory): [sigma(n^2)$n=1..50]; # _Muniru A Asiru_, Jan 01 2019
%t Table[Plus@@Divisors[n^2], {n, 48}] (* _Alonso del Arte_, Feb 24 2012 *)
%t f[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Sep 10 2020 *)
%o (MuPAD) numlib::sigma(n^2)$ n=1..81 // _Zerinvary Lajos_, May 13 2008
%o (Sage) [sigma(n^2,1)for n in range(1,49)] # _Zerinvary Lajos_, Jun 13 2009
%o (PARI) a(n) = sigma(n^2); \\ _Harry J. Smith_, Oct 30 2009
%o (Magma) [SumOfDivisors(n^2): n in [1..48]]; // _Bruno Berselli_, Apr 12 2011
%o (GAP) a:=List([1..50],n->Sigma(n^2));; Print(a); # _Muniru A Asiru_, Jan 01 2019
%o (Python)
%o from math import prod
%o from sympy import factorint
%o 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
%Y Cf. A000203, A000290, A028982, A319526.
%K nonn,easy,mult
%O 1,2
%A _Labos Elemer_, Nov 19 2001
|
