cp's OEIS Frontend

A361179 a(n) = sigma(n)^4.

%I A361179 #7 Mar 03 2023 06:27:46
%S A361179 1,81,256,2401,1296,20736,4096,50625,28561,104976,20736,614656,38416,
%T A361179 331776,331776,923521,104976,2313441,160000,3111696,1048576,1679616,
%U A361179 331776,12960000,923521,3111696,2560000,9834496,810000,26873856,1048576,15752961,5308416
%N A361179 a(n) = sigma(n)^4.
%C A361179 In general, for k>=1, Sum_{m=1..n} sigma(m)^k ~ c(k) * z(k) * n^(k+1) / (k+1), where z(k) = Product_{j=2..k+1} zeta(j).
%C A361179 z(k) tends to A021002 = 2.29485659167331379418351583... if k tends to infinity.
%C A361179 Table of logarithms of the first twenty constants c(k):
%C A361179 log(c1)  = 0
%C A361179 log(c2)  = 0.4185904294034097177091498674425959208785022862606440306200960821...
%C A361179 log(c3)  = 1.0423888168104400391462790418324165821902123159643681963298587386...
%C A361179 log(c4)  = 1.7991790110714031081639242851527957388041981665455193670488985855...
%C A361179 log(c5)  = 2.6531418047626712704435945717713008165192112256395129469527055461...
%C A361179 log(c6)  = 3.5826667694785981489341382260447390026333883927530294731356708082...
%C A361179 log(c7)  = 4.5733843557245275039380976990636718508529417039225677910093512418...
%C A361179 log(c8)  = 5.6152065176325962438798772352645945078887296036246579568363264836...
%C A361179 log(c9)  = 6.7007695219862872061684609152917692899880931107656334442026270254...
%C A361179 log(c10) = 7.8245175718301572361518558972457980392624870372412384620464547480...
%C A361179 log(c11) = 8.9821318589248960303876549202030018215854310738197659104984082438...
%C A361179 log(c12) = 10.170161510396427442300796140752106239603402200741405656518889304...
%C A361179 log(c13) = 11.385778844373902103940190311048453116470874526205115584130363228...
%C A361179 log(c14) = 12.626614423444098003503814842580453502016287945932183786430620101...
%C A361179 log(c15) = 13.890644760144907314506933347339629337810929043024214330654043796...
%C A361179 log(c16) = 15.176115136560648867246990011975416479066956527530401883224856531...
%C A361179 log(c17) = 16.481485806132270823150284520463000397265757050340939883069076823...
%C A361179 log(c18) = 17.805393674783928883671133007206209125657866860089528876021281793...
%C A361179 log(c19) = 19.146624201995507049618714377273936711664382470319966849198205155...
%C A361179 log(c20) = 20.504090088752226662590920186246482636058069128320785639131816842...
%C A361179 c1 = 1, c2 = 5/(2*zeta(2)) = 15/Pi^2.
%F A361179 Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^4.
%F A361179 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * zeta(s-4) * Product_{primes p} (1 + 1/p^(3*s-6) + 3/p^(2*s-3) + 5/p^(2*s-4) + 3/p^(2*s-5) + 3/p^(s-1) + 5/p^(s-2) + 3/p^(s-3)).
%F A361179 Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * zeta(5) * n^5 / 2700, where c = Product_{primes p} (1 + 3/p^2 + 5/p^3 + 3/p^4 + 3/p^5 + 5/p^6 + 3/p^7 + 1/p^9) = 6.0446828090651437986928739783339791032197283386377841627594461874871547391...
%F A361179 a(n) = A000583(A000203(n)).
%t A361179 Table[DivisorSigma[1, n]^4, {n, 1, 50}]
%o A361179 (PARI) a(n) = sigma(n)^4;
%o A361179 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X)*(1 + 3*p*X + 4*p^2*X + 3*p^3*X + p^4*X^2)/((1 - X)*(1 - p*X)*(1 - p^2*X)*(1 - p^3*X)*(1 - p^4*X)))[n], ", "))
%Y A361179 Cf. A000203, A072861, A361147.
%Y A361179 Cf. A361132, A361148.
%K A361179 nonn,mult
%O A361179 1,2
%A A361179 _Vaclav Kotesovec_, Mar 03 2023