A203557 -id:A203557 - 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

A203556 a(n) = sigma(n^5).

Original entry on oeis.org

1, 63, 364, 2047, 3906, 22932, 19608, 65535, 88573, 246078, 177156, 745108, 402234, 1235304, 1421784, 2097151, 1508598, 5580099, 2613660, 7995582, 7137312, 11160828, 6728904, 23854740, 12207031, 25340742, 21523360, 40137576, 21243690, 89572392, 29583456, 67108863

Offset: 1

Paul D. Hanna, Jan 03 2012

a(n) modulo 6 begins: [1,3,4,1,0,0,0,3,1,0,0,4,0,0,0,1,0,3,0,0,0,0,0,0,1,0,...], in which positions of nonzero residues seem related to squares.

			L.g.f.: L(x) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...
where the g.f. of A203557 begins:
exp(L(x)) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A203557 (exp), A000203 (sigma), A000584, A013664.

Variants: A065764, A175926, A202994.

f[p_, e_] := (p^(5*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 09 2020 *)
DivisorSigma[1,Range[40]^5] (* Harvey P. Dale, Dec 05 2021 *)

PARI
```
a(n) = sigma(n^5)
```

Logarithmic derivative of A203557.
Multiplicative with a(p^e) = (p^(5*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018
From Amiram Eldar, Nov 05 2022: (Start)
a(n) = A000203(A000584(n)) = A000203(n^5).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3220880186... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

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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A203556 a(n) = sigma(n^5).

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

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