A084218 - OEIS

%I #32 May 18 2024 11:35:11

%S 1,13,73,205,601,949,2353,3277,5905,7813,14521,14965,28393,30589,

%T 43873,52429,83233,76765,129961,123205,171769,188773,279313,239221,

%U 375601,369109,478297,482365,706441,570349,922561,838861,1060033,1082029

%N a(n) = sigma_4(n^2)/sigma_2(n^2).

%H G. C. Greubel, <a href="/A084218/b084218.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - _Amiram Eldar_, Sep 13 2020

%F Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - _Vaclav Kotesovec_, Sep 24 2020

%F Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - _Amiram Eldar_, Oct 30 2022

%F From _Peter Bala_, Jan 18 2024: (Start)

%F 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 a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.

%F Dirichlet g.f.: zeta(s-4)/zeta(s-2). (End)

%F a(n) = Sum_{d|n} mu(n/d) * (n/d)^2* sigma_4(d). - _Seiichi Manyama_, May 18 2024

%p 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 Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* _G. C. Greubel_, Oct 08 2018 *)

%t 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 (PARI) a(n)=sumdiv(n^2,d,d^4)/sumdiv(n^2,d,d^2)

%o (PARI) a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ _Michel Marcus_, Oct 09 2018

%Y Cf. A001157, A001159, A002117, A007434, A013663, A057660, A065827.

%K nonn,mult,easy

%O 1,2

%A _Benoit Cloitre_, Jun 21 2003