OFFSET
1,2
REFERENCES
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
FORMULA
Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
Logarithmic derivative of A224364. - Paul D. Hanna, Apr 04 2013
Bisection: a(2*k-1) = A001157(2*k-1), a(2*k) = 4*A001157(k) - A050999(2*k), k >= 1. In the Hardy reference a(n) = sigma^*_2(n). - Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{k>=1} k^2*x^k/(1 - (-x)^k). - Ilya Gutkovskiy, Nov 09 2018
Sum_{k=1..n} a(k) ~ 7 * zeta(3) * n^3 / 24. - Vaclav Kotesovec, Nov 10 2018
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 1/2^(s-1) + 1/2^(2*s-3)). - Amiram Eldar, Sep 21 2023
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + ...
where exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 + 106*x^8 + 181*x^9 + ... + A224364(n)*x^n + ... - Paul D. Hanna, Apr 04 2013
MATHEMATICA
a[n_] := (-1)^n DivisorSum[n, (-1)^# * #^2 &]; Array[a, 50] (* Jean-François Alcover, Dec 23 2015 *)
a[n_] := If[OddQ[n], 1, (1 - 6/(4^(IntegerExponent[n, 2] + 1) - 1))] * DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)
PROG
(PARI) {a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ Paul D. Hanna, Apr 04 2013
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
CROSSREFS
KEYWORD
mult,easy,nonn
AUTHOR
Vladeta Jovovic, Sep 11 2001
STATUS
approved