A034714 Dirichlet convolution of squares with themselves.
1, 8, 18, 48, 50, 144, 98, 256, 243, 400, 242, 864, 338, 784, 900, 1280, 578, 1944, 722, 2400, 1764, 1936, 1058, 4608, 1875, 2704, 2916, 4704, 1682, 7200, 1922, 6144, 4356, 4624, 4900, 11664, 2738, 5776, 6084, 12800, 3362, 14112, 3698, 11616, 12150, 8464
Offset: 1
Links
- Bruno Berselli, Table of n, a(n) for n = 1..1000
- Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Programs
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Magma
[n^2*NumberOfDivisors(n): n in [1..50]]; // Bruno Berselli, Feb 12 2014
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Maple
A034714 := proc(n) n^2*numtheory[tau](n) ; end proc: seq(A034714(n),n=1..20) ; # R. J. Mathar, Feb 03 2011
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Mathematica
A034714[n_]:=DivisorSigma[0,n]*n^2; Array[A034714, 50] (* Enrique Pérez Herrero, Jun 26 2011 *)
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PARI
A034714(n)=numdiv(n)*n^2 \\ Enrique Pérez Herrero, Jun 26 2011
Formula
Dirichlet g.f.: zeta^2(s-2).
Multiplicative with a(p^e) = (e+1)p^(2e). - Mitch Harris, Jun 27 2005
Row sums of triangle A134576. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} k^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = n * A038040(n). - Torlach Rush, Feb 01 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (-p^2 * log(1 - 1/p^2)) = 1.27728092754165872535305748273941301416624226497497308879403022758421224... - Vaclav Kotesovec, Sep 19 2020
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
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
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