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

1, 81, 256, 2401, 1296, 20736, 4096, 50625, 28561, 104976, 20736, 614656, 38416, 331776, 331776, 923521, 104976, 2313441, 160000, 3111696, 1048576, 1679616, 331776, 12960000, 923521, 3111696, 2560000, 9834496, 810000, 26873856, 1048576, 15752961, 5308416

Offset: 1

Author

Vaclav Kotesovec, Mar 03 2023

Keywords

• nonn
• mult

Comments

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).

z(k) tends to A021002 = 2.29485659167331379418351583... if k tends to infinity.

Table of logarithms of the first twenty constants c(k):

log(c1) = 0

log(c2) = 0.4185904294034097177091498674425959208785022862606440306200960821...

log(c3) = 1.0423888168104400391462790418324165821902123159643681963298587386...

log(c4) = 1.7991790110714031081639242851527957388041981665455193670488985855...

log(c5) = 2.6531418047626712704435945717713008165192112256395129469527055461...

log(c6) = 3.5826667694785981489341382260447390026333883927530294731356708082...

log(c7) = 4.5733843557245275039380976990636718508529417039225677910093512418...

log(c8) = 5.6152065176325962438798772352645945078887296036246579568363264836...

log(c9) = 6.7007695219862872061684609152917692899880931107656334442026270254...

log(c10) = 7.8245175718301572361518558972457980392624870372412384620464547480...

log(c11) = 8.9821318589248960303876549202030018215854310738197659104984082438...

log(c12) = 10.170161510396427442300796140752106239603402200741405656518889304...

log(c13) = 11.385778844373902103940190311048453116470874526205115584130363228...

log(c14) = 12.626614423444098003503814842580453502016287945932183786430620101...

log(c15) = 13.890644760144907314506933347339629337810929043024214330654043796...

log(c16) = 15.176115136560648867246990011975416479066956527530401883224856531...

log(c17) = 16.481485806132270823150284520463000397265757050340939883069076823...

log(c18) = 17.805393674783928883671133007206209125657866860089528876021281793...

log(c19) = 19.146624201995507049618714377273936711664382470319966849198205155...

log(c20) = 20.504090088752226662590920186246482636058069128320785639131816842...

c1 = 1, c2 = 5/(2*zeta(2)) = 15/Pi^2.

Crossrefs

Cf. A000203, A072861, A361147.

Cf. A361132, A361148.

Programs

Mathematica

Table[DivisorSigma[1, n]^4, {n, 1, 50}]

PARI

a(n) = sigma(n)^4;

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], ", "))

Formula

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^4.

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)).

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...

a(n) = A000583(A000203(n)).