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A202994 a(n) = sigma(n^4).

%I A202994 #31 Oct 25 2023 14:27:58
%S A202994 1,31,121,511,781,3751,2801,8191,9841,24211,16105,61831,30941,86831,
%T A202994 94501,131071,88741,305071,137561,399091,338921,499255,292561,991111,
%U A202994 488281,959171,797161,1431311,732541,2929531,954305,2097151,1948705,2750971,2187581
%N A202994 a(n) = sigma(n^4).
%C A202994 Here sigma(n^4) denotes the sums of divisors of n^4.
%H A202994 T. D. Noe, <a href="/A202994/b202994.txt">Table of n, a(n) for n = 1..1000</a>
%F A202994 a(11*n) == 0 (mod 5) iff gcd(n,11) = 1.
%F A202994 Logarithmic derivative of A202993.
%F A202994 Multiplicative with a(p^e) = (p^(4*e+1)-1)/(p-1) for prime p. - _Andrew Howroyd_, Jul 23 2018
%F A202994 a(n) = A000203(A000583(n)). - _Michel Marcus_, Sep 10 2020
%F A202994 Sum_{k>=1} 1/a(k) = 1.04483665108279017775482622699860068916340892303889072390102812885655694752... - _Vaclav Kotesovec_, Sep 20 2020
%F A202994 Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.3840585791... . - _Amiram Eldar_, Nov 05 2022
%e A202994 L.g.f.: L(x) = x + 31*x^2/2 + 121*x^3/3 + 511*x^4/4 + 781*x^5/5 + 3751*x^6/6 +...
%e A202994 where exp(L(x)) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...+ A202993(n)*x^n +...
%t A202994 DivisorSigma[1,Range[40]^4] (* _Harvey P. Dale_, Jan 29 2012 *)
%t A202994 f[p_, e_] := (p^(4*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Sep 10 2020 *)
%o A202994 (PARI) {a(n)=sigma(n^4)}
%o A202994 (Python)
%o A202994 from math import prod
%o A202994 from sympy import factorint
%o A202994 def A202994(n): return prod((p**((e<<2)+1)-1)//(p-1) for p,e in factorint(n).items()) # _Chai Wah Wu_, Oct 25 2023
%Y A202994 Cf. A000203 (sigma), A000583, A013663, A065764, A175926, A202993.
%K A202994 nonn,mult
%O A202994 1,2
%A A202994 _Paul D. Hanna_, Dec 27 2011
%E A202994 Keyword:mult added by _Andrew Howroyd_, Jul 23 2018