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 A351314 #27 Jul 19 2025 13:11:31
%S A351314 1,1,1,65537,1,1,1,65537,43046722,1,1,65537,1,1,1,4295032833,1,
%T A351314 43046722,1,65537,1,1,1,65537,152587890626,1,43046722,65537,1,1,1,
%U A351314 4295032833,1,1,1,2821153019714,1,1,1,65537,1,1,1,65537,43046722,1,1,4295032833,33232930569602,152587890626
%N A351314 Sum of the 8th powers of the square divisors of n.
%C A351314 Inverse Möbius transform of n^8 * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 21 2024
%H A351314 Seiichi Manyama, <a href="/A351314/b351314.txt">Table of n, a(n) for n = 1..10000</a>
%F A351314 a(n) = Sum_{d^2|n} (d^2)^8.
%F A351314 Multiplicative with a(p) = (p^(16*(1+floor(e/2))) - 1)/(p^16 - 1). - _Amiram Eldar_, Feb 07 2022
%F A351314 G.f.: Sum_{k>0} k^16*x^(k^2)/(1-x^(k^2)). - _Seiichi Manyama_, Feb 12 2022
%F A351314 From _Amiram Eldar_, Sep 20 2023: (Start)
%F A351314 Dirichlet g.f.: zeta(s) * zeta(2*s-16).
%F A351314 Sum_{k=1..n} a(k) ~ (zeta(17/2)/17) * n^(17/2). (End)
%F A351314 a(n) = Sum_{d|n} d^8 * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 21 2024
%F A351314 a(n) = Sum_{d|n} lambda(d)*d^8*sigma_8(n/d), where lambda = A008836. - _Ridouane Oudra_, Jul 19 2025
%e A351314 a(16) = 4295032833; a(16) = Sum_{d^2|16} (d^2)^8 = (1^2)^8 + (2^2)^8 + (4^2)^8 = 4295032833.
%t A351314 f[p_, e_] := (p^(16*(1 + Floor[e/2])) - 1)/(p^16 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 07 2022 *)
%t A351314 Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^8],{n,80}] (* _Harvey P. Dale_, Feb 13 2022 *)
%o A351314 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^16*x^k^2/(1-x^k^2))) \\ _Seiichi Manyama_, Feb 12 2022
%Y A351314 Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), this sequence (k=8), A351315 (k=9), A351316 (k=10).
%Y A351314 Cf. A010052, A008836, A013956.
%K A351314 nonn,easy,mult
%O A351314 1,4
%A A351314 _Wesley Ivan Hurt_, Feb 06 2022