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

1,2

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).

Formula

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

Equals n^2*tau(n), where tau(n) = A000005(n) = number of divisors of n. - Jon Perry, Aug 28 2005

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

Maple

A034714 := proc(n) n^2*numtheory[tau](n) ; end proc:
seq(A034714(n), n=1..20) ; # R. J. Mathar, Feb 03 2011

Mathematica

A034714[n_]:=DivisorSigma[0, n]*n^2; Array[A034714, 50] (* Enrique Pérez Herrero, Jun 26 2011 *)

Prog

(PARI) A034714(n)=numdiv(n)*n^2 \\ Enrique Pérez Herrero, Jun 26 2011
(Magma) [n^2*NumberOfDivisors(n): n in [1..50]]; // Bruno Berselli, Feb 12 2014

Crossrefs

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

Sequence in context: A395591 A192311 A300161 * A153388 A109988 A335440

Adjacent sequences: A034711 A034712 A034713 * A034715 A034716 A034717

Keyword

nonn,easy,mult

Author

Erich Friedman

Status

approved