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

%I A328667 #16 Jun 19 2025 09:19:31
%S A328667 1,5,10,13,26,50,50,45,91,130,122,130,170,250,260,173,290,455,362,338,
%T A328667 500,610,530,450,651,850,820,650,842,1300,962,685,1220,1450,1300,1183,
%U A328667 1370,1810,1700,1170,1682,2500,1850,1586,2366,2650,2210,1730,2451,3255
%N A328667 a(n) = Sum_{d divides n} (-1)^(n + 1 + d + n/d) * d^2.
%H A328667 Seiichi Manyama, <a href="/A328667/b328667.txt">Table of n, a(n) for n = 1..10000</a>
%F A328667 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).
%F A328667 G.f.: Sum_{k>=1} k^2 * x^k/(1 + (-x)^k) = Sum_{k>=1} x^k*(1 - (-x)^k)/(1 + (-x)^k)^3.
%F A328667 a(n) = -(-1)^n*A321558(n). a(2*n - 1) = A001157(2*n - 1) = A099978(n). a(4*n + 2) = A001157(4*n + 2).
%F A328667 Sum_{k=1..n} a(k) ~ c * n^3, where c = 7*zeta(3)/24 = 0.350599... . - _Amiram Eldar_, Nov 01 2022
%e A328667 G.f. = x + 5*x^2 + 10*x^3 + 13*x^4 + 26*x^5 + 50*x^6 + 50*x^7 + 45*x^8 + ...
%t A328667 a[ n_] := If[ n < 1, 0, DivisorSum[n, (-1)^(n + 1 + # + n/#) #^2 &]];
%o A328667 (PARI) {a(n) = sumdiv(n, d, (-1)^(n + 1 + n\d + d)*d^2)};
%Y A328667 Cf. A001157, A002117, A099978, A321358, A321558.
%K A328667 nonn,mult
%O A328667 1,2
%A A328667 _Michael Somos_, Oct 24 2019