A064027 a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.

1, 3, 10, 19, 26, 30, 50, 83, 91, 78, 122, 190, 170, 150, 260, 339, 290, 273, 362, 494, 500, 366, 530, 830, 651, 510, 820, 950, 842, 780, 962, 1363, 1220, 870, 1300, 1729, 1370, 1086, 1700, 2158, 1682, 1500, 1850, 2318, 2366, 1590, 2210, 3390, 2451, 1953

Offset: 1

Author

Vladeta Jovovic, Sep 11 2001

Keywords

• mult
• easy
• nonn

Examples

			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
		

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.

Crossrefs

Cf. A001157, A002129, A008457, A050999, A224364.

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

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 *)

PARI

{a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ Paul D. Hanna, Apr 04 2013

Formula

Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.

a(n) = (-1)^n*(A001157(n) - 2*A050999(n)).

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