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 A358336 #17 Dec 13 2024 10:25:16
%S A358336 1,3,5,12,9,15,13,40,30,27,21,60,25,39,45,120,33,90,37,108,65,63,45,
%T A358336 200,90,75,153,156,57,135,61,336,105,99,117,360,73,111,125,360,81,195,
%U A358336 85,252,270,135,93,600,182,270,165,300,105,459,189,520,185,171,117,540,121,183,390,896
%N A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1) for prime p and e > 0.
%F A358336 a(n) = Sum_{k=1..n} gcd(k, n) * A005361(gcd(k, n)) for n > 0.
%F A358336 Equals Dirichlet convolution of A000010 and n * A005361.
%F A358336 Dirichlet g.f.: (zeta(s-1)^2 * zeta(2*s-2) * zeta(3*s-3)) / (zeta(s) * zeta(6*s-6)).
%F A358336 Equals Dirichlet convolution of A018804 and A112526.
%F A358336 Sum_{k=1..n} a(k) ~ (zeta(3)/(2*zeta(6))) * n^2 * (log(n) + 2*gamma - 1/2 + zeta'(2)/zeta(2) + 3*zeta'(3)/zeta(3) + 6*zeta'(6)/zeta(6)), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 13 2024
%t A358336 f[p_, e_] := ((p - 1)*(1 + e*(e + 1)/2) + e)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2022 *)
%o A358336 (PARI) a(n) = { my (f=factor(n), p, e, v=1); for (k=1, #f~, p=f[k,1]; e=f[k,2]; v *= ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1)); return (v) } \\ _Rémy Sigrist_, Jan 18 2023
%Y A358336 Cf. A000010, A005361, A018804, A112526.
%Y A358336 Cf. A001620, A002117, A013664, A244115, A157289, A306016.
%K A358336 nonn,easy,mult
%O A358336 1,2
%A A358336 _Werner Schulte_, Nov 09 2022