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Dirichlet convolution of squares with themselves.
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%I #55 Jan 22 2024 08:56:55

%S 1,8,18,48,50,144,98,256,243,400,242,864,338,784,900,1280,578,1944,

%T 722,2400,1764,1936,1058,4608,1875,2704,2916,4704,1682,7200,1922,6144,

%U 4356,4624,4900,11664,2738,5776,6084,12800,3362,14112,3698,11616,12150,8464

%N Dirichlet convolution of squares with themselves.

%H Bruno Berselli, <a href="/A034714/b034714.txt">Table of n, a(n) for n = 1..1000</a>

%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).

%F Dirichlet g.f.: zeta^2(s-2).

%F Equals n^2*tau(n), where tau(n) = A000005(n) = number of divisors of n. - _Jon Perry_, Aug 28 2005

%F Multiplicative with a(p^e) = (e+1)p^(2e). - _Mitch Harris_, Jun 27 2005

%F Row sums of triangle A134576. - _Gary W. Adamson_, Nov 02 2007

%F G.f.: Sum_{k>=1} k^2*x^k*(1 + x^k)/(1 - x^k)^3. - _Ilya Gutkovskiy_, Oct 24 2018

%F a(n) = n * A038040(n). - _Torlach Rush_, Feb 01 2019

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (-p^2 * log(1 - 1/p^2)) = 1.27728092754165872535305748273941301416624226497497308879403022758421224... - _Vaclav Kotesovec_, Sep 19 2020

%F G.f.: Sum_{n >= 1} q^(n^2)*( n^4*q^(3*n) - n^2*(n^2 + 4*n - 2)*q^(2*n) - n^2*(n^2 - 4*n - 2)*q^n + n^4 )/(1 - q^n)^3 - apply the operator q*d/dq twice to equation 5 in Arndt and set x = 1. - _Peter Bala_, Jan 21 2021

%F Sum_{k=1..n} a(k) ~ (n^3/3) * (log(n) + 2*gamma - 1/3), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Nov 02 2023

%F a(n) = Sum_{1 <= i, j <= n} sigma_2( gcd(i, j, n) ) = Sum_{d divides n} sigma_2(d) * J_2(n/d), where sigma_2(n) = A001157(n) and the Jordan totient function J_2(n) = A007434(n). - _Peter Bala_, Jan 22 2024

%p A034714 := proc(n) n^2*numtheory[tau](n) ; end proc:

%p seq(A034714(n),n=1..20) ; # _R. J. Mathar_, Feb 03 2011

%t A034714[n_]:=DivisorSigma[0,n]*n^2; Array[A034714, 50] (* _Enrique Pérez Herrero_, Jun 26 2011 *)

%o (PARI) A034714(n)=numdiv(n)*n^2 \\ _Enrique Pérez Herrero_, Jun 26 2011

%o (Magma) [n^2*NumberOfDivisors(n): n in [1..50]]; // _Bruno Berselli_, Feb 12 2014

%Y Cf. A000005, A000290, A001620, A038040, A134576, A319085 (partial sums).

%K nonn,easy,mult

%O 1,2

%A _Erich Friedman_