A202993 -id:A202993 - cp's OEIS frontend

A156304 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.

1, 1, 8, 21, 77, 199, 661, 1663, 4852, 12382, 33289, 82877, 213026, 518109, 1279852, 3053404, 7312985, 17093793, 39952528, 91661695, 209709116, 473095589, 1062567288, 2359804486, 5214774263, 11415904502, 24860918943, 53709881911

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Euler transform of A160889. - Vaclav Kotesovec, Nov 01 2024

			G.f.: A(x) = 1 + x + 8*x^2 + 21*x^3 + 77*x^4 + 199*x^5 + 661*x^6 +...
log(A(x)) = x + 15*x^2/2 + 40*x^3/3 + 127*x^4/4 + 156*x^5/5 + 600*x^6/6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000203 (sigma), A000041 (partitions), A156303, A202993, A203557.

Cf. A160889, A175926, A330595.

{a(n)=polcoeff(exp(sum(m=1,n,sigma(m^3)*x^m/m)+x*O(x^n)),n)}

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k^3)*a(n-k)))}

a(n) = (1/n)*Sum_{k=1..n} sigma(k^3) * a(n-k) for n>0, with a(0)=1.

log(a(n)) ~ 4*Pi*c^(1/4)*n^(3/4) / (3^(5/4)*5^(1/4)), where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699... - Vaclav Kotesovec, Nov 01 2024

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

Original entry on oeis.org

1, 31, 121, 511, 781, 3751, 2801, 8191, 9841, 24211, 16105, 61831, 30941, 86831, 94501, 131071, 88741, 305071, 137561, 399091, 338921, 499255, 292561, 991111, 488281, 959171, 797161, 1431311, 732541, 2929531, 954305, 2097151, 1948705, 2750971, 2187581

Offset: 1

Paul D. Hanna, Dec 27 2011

Here sigma(n^4) denotes the sums of divisors of n^4.

			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 +...
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. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000203 (sigma), A000583, A013663, A065764, A175926, A202993.

DivisorSigma[1,Range[40]^4] (* Harvey P. Dale, Jan 29 2012 *)
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 *)

PARI
```
{a(n)=sigma(n^4)}
```

from math import prod
from sympy import factorint
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

a(11*n) == 0 (mod 5) iff gcd(n,11) = 1.
Logarithmic derivative of A202993.
Multiplicative with a(p^e) = (p^(4*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018
a(n) = A000203(A000583(n)). - Michel Marcus, Sep 10 2020
Sum_{k>=1} 1/a(k) = 1.04483665108279017775482622699860068916340892303889072390102812885655694752... - Vaclav Kotesovec, Sep 20 2020
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

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A203557 G.f.: exp( Sum_{n>=1} sigma(n^5)*x^n/n ).

Original entry on oeis.org

1, 1, 32, 153, 1145, 5677, 37641, 184685, 1047862, 5196410, 26935148, 129702476, 638028933, 2987297287, 14055935617, 64139004752, 291595380989, 1296984485909, 5732084828019, 24910785830408, 107411267744602, 457008372687439, 1928413165110846, 8046605441623654

Offset: 0

Paul D. Hanna, Jan 03 2012

nonn

In general, if m >= 1 and g.f.= exp(Sum_{k>=1} sigma(k^m)*x^k/k), then log(a(n)) ~ (1 + 1/m) * (c*m!)^(1/(m+1)) * n^(m/(m+1)), where c = Product_{primes p} ((p^(m+2) - p^(m+1) + p^m - p) / ((p-1)*(p^(m+1)-1))). - Vaclav Kotesovec, Nov 01 2024

			G.f.: A(x) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...
where the logarithm equals the l.g.f. of A203556:
log(A(x)) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...+ sigma(n^5)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A203556, A000203 (sigma); variants: A000041 (m=1), A156303 (m=2), A156304 (m=3), A202993 (m=4).

{a(n)=polcoeff(exp(sum(m=1,n,sigma(m^5)*x^m/m)+x*O(x^n)),n)}

Logarithmic derivative yields A203556.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A203556(k)*a(n-k) for n > 0. - Seiichi Manyama, Sep 09 2020
log(a(n)) ~ 2^(3/2) * 3^(7/6) * c^(1/6) * n^(5/6) / 5^(5/6), where c = Product_{primes p} (p*(1 + p + p^2 + p^3 + p^5) / (p^6 - 1)) = 1.93252811194652723494722635658171746713... - Vaclav Kotesovec, Nov 01 2024

A319363 a(n) = [x^n] exp(Sum_{k>=1} sigma(k^n)*x^k/k).

Original entry on oeis.org

1, 1, 4, 21, 296, 5677, 291348, 22679763, 3946629792, 1281287791090, 840017139222515, 1068253745514088673, 2745322115165570881474, 14006438682727577999625359, 141884380264686466724199980066, 2897075017978846591982107951045864, 118770312781918226439371316982748736112

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A000041, A156303, A156304, A202993, A203557, A319362.

Table[SeriesCoefficient[Exp[Sum[DivisorSigma[1, k^n] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

cp's OEIS Frontend

A156304 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.

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

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A203557 G.f.: exp( Sum_{n>=1} sigma(n^5)*x^n/n ).

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PARI

Formula

A319363 a(n) = [x^n] exp(Sum_{k>=1} sigma(k^n)*x^k/k).

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Mathematica