This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A327242 #23 May 26 2025 09:54:52
%S A327242 1,0,5,-5,7,0,9,-18,18,0,13,-25,15,0,35,-47,19,0,21,-35,45,0,25,-90,
%T A327242 38,0,58,-45,31,0,33,-108,65,0,63,-90,39,0,75,-126,43,0,45,-65,126,0,
%U A327242 49,-235,66,0,95,-75,55,0,91,-162,105,0,61,-175,63,0,162,-233,105
%N A327242 Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.
%C A327242 Inverse Moebius transform of A002129.
%C A327242 Dirichlet convolution of A000005 with A181983.
%H A327242 Amiram Eldar, <a href="/A327242/b327242.txt">Table of n, a(n) for n = 1..10000</a>
%F A327242 a(n) = Sum_{d|n} A002129(d).
%F A327242 a(n) = Sum_{d|n} (-1)^(d + 1) * d * tau(n/d).
%F A327242 Multiplicative with a(2^e) = 3*e + 5 - 2^(e+2), and a(p^e) = (p^(e+2) - (e+2)*p +e + 1)/(p-1)^2 for an odd prime p. - _Amiram Eldar_, May 25 2025
%t A327242 nmax = 65; CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A327242 a[n_] := Sum[(-1)^(d + 1) d DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
%t A327242 f[p_, e_] := (p^(e + 2) - (e + 2)*p + e + 1)/(p-1)^2; f[2, e_] := 3*e + 5 - 2^(e+2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 25 2025 *)
%o A327242 (PARI) a(n) = {sumdiv(n, d, (-1)^(d + 1) * d * numdiv(n/d))} \\ _Andrew Howroyd_, Sep 14 2019
%o A327242 (Magma) [&+[(-1)^(d+1)*d*#Divisors(n div d):d in Divisors(n)]:n in [1..65]]; // _Marius A. Burtea_, Sep 14 2019
%Y A327242 Cf. A000005, A002129, A007429, A008586 (positions of negative terms), A016825 (positions of 0's), A181983, A288417, A288571.
%K A327242 sign,mult
%O A327242 1,3
%A A327242 _Ilya Gutkovskiy_, Sep 14 2019