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 A318845 #11 May 27 2025 01:08:54
%S A318845 1,3,6,6,8,18,10,10,24,24,14,36,16,30,48,15,20,72,22,48,60,42,26,60,
%T A318845 46,48,82,60,32,144,34,21,84,60,80,144,40,66,96,80,44,180,46,84,192,
%U A318845 78,50,90,76,138,120,96,56,246,112,100,132,96,62,288,64,102,240,28,128,252,70,120,156,240
%N A318845 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} sigma(j), where sigma(j) = sum of divisors of j (A000203).
%H A318845 Amiram Eldar, <a href="/A318845/b318845.txt">Table of n, a(n) for n = 1..10000</a>
%F A318845 G.f.: Sum_{k>=1} A007429(k)*x^k/(1 + x^k).
%F A318845 L.g.f.: log(Product_{k>=1} (1 + x^k)^(A007429(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
%F A318845 From _Amiram Eldar_, May 26 2025: (Start)
%F A318845 Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3) for an odd prime p.
%F A318845 Dirichlet g.f: zeta(s-1) * zeta(s)^2 * (1 - 1/2^(s-1)).
%F A318845 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^6/864 = 1.112718... . (End)
%t A318845 Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[1, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
%t A318845 nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[1, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A318845 nmax = 70; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[1, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t A318845 f[p_, e_] := (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3); f[2, e_] := (e+1)*(e+2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2025 *)
%o A318845 (PARI) a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, (e+1)*(e+2)/2, (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3)));} \\ _Amiram Eldar_, May 26 2025
%Y A318845 Cf. A000203, A007429, A007430, A288417, A318768.
%K A318845 nonn,mult
%O A318845 1,2
%A A318845 _Ilya Gutkovskiy_, Sep 04 2018