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 A084218 #46 May 21 2025 01:34:06
%S A084218 1,13,73,205,601,949,2353,3277,5905,7813,14521,14965,28393,30589,
%T A084218 43873,52429,83233,76765,129961,123205,171769,188773,279313,239221,
%U A084218 375601,369109,478297,482365,706441,570349,922561,838861,1060033,1082029
%N A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).
%H A084218 G. C. Greubel, <a href="/A084218/b084218.txt">Table of n, a(n) for n = 1..10000</a>
%F A084218 Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - _Amiram Eldar_, Sep 13 2020
%F A084218 Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - _Vaclav Kotesovec_, Sep 24 2020
%F A084218 Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - _Amiram Eldar_, Oct 30 2022
%F A084218 From _Peter Bala_, Jan 18 2024: (Start)
%F A084218 a(n) = Sum_{d divides n} J_2(d^2) = Sum_{d divides n} d^2 * J_2(d), where the Jordan totient function J_2(n) = A007434(n).
%F A084218 a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.
%F A084218 Dirichlet g.f.: zeta(s) * zeta(s-4) / zeta(s-2) [Corrected by _Michael Shamos_, May 18 2025]. (End)
%F A084218 a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_4(d). - _Seiichi Manyama_, May 18 2024
%p A084218 with(numtheory): a:=n->sigma[4](n^2)/sigma[2](n^2): seq(a(n),n=1..40); # _Muniru A Asiru_, Oct 09 2018
%t A084218 Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* _G. C. Greubel_, Oct 08 2018 *)
%t A084218 f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* _Amiram Eldar_, Sep 13 2020 *)
%o A084218 (PARI) a(n)=sumdiv(n^2,d,d^4)/sumdiv(n^2,d,d^2)
%o A084218 (PARI) a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ _Michel Marcus_, Oct 09 2018
%Y A084218 Cf. A001157, A001159, A002117, A007434, A013663, A065827.
%Y A084218 Cf. A068963, A372962, A372963, A373007.
%Y A084218 Cf. A057660, A084220, A372966, A373105.
%K A084218 nonn,mult,easy
%O A084218 1,2
%A A084218 _Benoit Cloitre_, Jun 21 2003