A328667 a(n) = Sum_{d divides n} (-1)^(n + 1 + d + n/d) * d^2.

1, 5, 10, 13, 26, 50, 50, 45, 91, 130, 122, 130, 170, 250, 260, 173, 290, 455, 362, 338, 500, 610, 530, 450, 651, 850, 820, 650, 842, 1300, 962, 685, 1220, 1450, 1300, 1183, 1370, 1810, 1700, 1170, 1682, 2500, 1850, 1586, 2366, 2650, 2210, 1730, 2451, 3255

Offset: 1

Author

Michael Somos, Oct 24 2019

Keywords

• nonn
• mult

Examples

			G.f. = x + 5*x^2 + 10*x^3 + 13*x^4 + 26*x^5 + 50*x^6 + 50*x^7 + 45*x^8 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001157, A002117, A099978, A321358, A321558.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[n, (-1)^(n + 1 + # + n/#) #^2 &]];

PARI

{a(n) = sumdiv(n, d, (-1)^(n + 1 + n\d + d)*d^2)};

Formula

Multiplicative with a(2^e) = (2^(2*e+1) + 7)/3 = A321358(e) if e>0, else a(p^e) = (p^(2*e+2) - 1)/(p^2 - 1).

G.f.: Sum_{k>=1} k^2 * x^k/(1 + (-x)^k) = Sum_{k>=1} x^k*(1 - (-x)^k)/(1 + (-x)^k)^3.

a(n) = -(-1)^n*A321558(n). a(2*n - 1) = A001157(2*n - 1) = A099978(n). a(4*n + 2) = A001157(4*n + 2).

Sum_{k=1..n} a(k) ~ c * n^3, where c = 7*zeta(3)/24 = 0.350599... . - Amiram Eldar, Nov 01 2022