cp's OEIS Frontend

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.

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A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

Original entry on oeis.org

1, 1, 14, 35, 205, 521, 2507, 6709, 26712, 73834, 262431, 724537, 2384988, 6552033, 20289864, 55244988, 163342701, 439201501, 1251532060, 3321188863, 9177476977, 24028568664, 64709650590, 167153761523, 440300702427, 1122562426240, 2900254892900, 7301575351055, 18544013542057
Offset: 0

Views

Author

Joerg Arndt, Dec 30 2010

Keywords

Comments

Compare with g.f. for partition numbers A000041: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Similarly, exp( Sum_{n>=1} sigma(n)^2*x^n/n ) gives A156302.

Examples

			G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...
such that, by definition,
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 6^3*x^5/5 + 12^3*x^6/6 +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A156302, A205797, A361147.

Programs

  • Mathematica
    nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1,k]^3 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 30 2024 *)
  • PARI
    N=100;v=Vec(exp(sum(k=1,N,sigma(k)^3*x^k/k)+x*O(x^N)))
  • PARI
    a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^3*a(n-k)))
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^2*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

Formula

a(0)=0 and a(n)=1/n*sum(k=1,n,sigma(k)^3*a(n-k)) for n>0.
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^2 * x^(n*k) / n ). [Paul D. Hanna, Jan 31 2012]
From Vaclav Kotesovec, Oct 30 2024: (Start)
log(a(n)) ~ 2^(7/4) * c^(1/4) * Pi^(3/2) * zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)), where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.8359835743341928677044245715803848964089818378791769798895797934086403174189...
Equivalently, log(a(n)) ~ 3.2753680082113515869730831738879060384726246... * n^(3/4). (End)

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

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 03 2023

Keywords

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)).
Showing 1-2 of 2 results.

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