A203556 -id:A203556 - cp's OEIS frontend

A175926 Sum of divisors of cubes.

1, 15, 40, 127, 156, 600, 400, 1023, 1093, 2340, 1464, 5080, 2380, 6000, 6240, 8191, 5220, 16395, 7240, 19812, 16000, 21960, 12720, 40920, 19531, 35700, 29524, 50800, 25260, 93600, 30784, 65535, 58560, 78300, 62400, 138811, 52060, 108600, 95200

Offset: 1

Zak Seidov, Oct 19 2010

The Mobius transform of the sequence is 1, 14, 39 ,112, 155,..., which equals the sequence defined by n*A160889(n). - R. J. Mathar, Apr 15 2011

Zhi-Wei Sun noted that the first 10^7 terms are pairwise distinct, but Noam D. Elkies found that a(48142241) = a(48374911), a(384422506) = a(403764207) and so on. - Zhi-Wei Sun, Jan 08 2014

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

Cf. sigma(n^k): A000203 (k=1), A065764 (k=2), this sequence (k=3), A202994 (k=4), A203556 (k=5).

Cf. A000578, A013662, A160889.

[ SumOfDivisors(n^3) : n in [1..100]]; // Vincenzo Librandi, Apr 14 2011

DivisorSigma[1,#]&/@((Range[40])^3) (* Harvey P. Dale, Aug 30 2015 *)
f[p_, e_] := (p^(3*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

a(n) = sigma(n^3); \\ Amiram Eldar, Nov 05 2022

from math import prod
from sympy import factorint
def A175926(n): return prod((p**(3*e+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

a(n) = A000203(n^3). - R. J. Mathar, Mar 31 2011
Multiplicative with a(p^e) = (p^(3e+1)-1)/(p-1). - R. J. Mathar, Mar 31 2011
Sum_{k>=1} 1/a(k) = 1.11535899887110289127674868460900333554265894187008102863022551119560512446... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = 0.4732277044... . - Amiram Eldar, Nov 05 2022

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

A344328 Number of divisors of n^5.

Original entry on oeis.org

1, 6, 6, 11, 6, 36, 6, 16, 11, 36, 6, 66, 6, 36, 36, 21, 6, 66, 6, 66, 36, 36, 6, 96, 11, 36, 16, 66, 6, 216, 6, 26, 36, 36, 36, 121, 6, 36, 36, 96, 6, 216, 6, 66, 66, 36, 6, 126, 11, 66, 36, 66, 6, 96, 36, 96, 36, 36, 6, 396, 6, 36, 66, 31, 36, 216, 6, 66, 36, 216, 6, 176, 6, 36, 66, 66, 36

Offset: 1

Seiichi Manyama, May 15 2021

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

Column k=5 of A343656.

Cf. A000005, A000584, A082476 (5^omega(n)), A203556.

Table[DivisorSigma[0, n^5], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^5);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 5*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 5^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 5^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A000584(n)).
Multiplicative with a(p^e) = 5*e+1.
a(n) = Sum_{d|n} 5^omega(d).
G.f.: Sum_{k>=1} 5^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 4/p^s). - Vaclav Kotesovec, Aug 19 2021

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A175926 Sum of divisors of cubes.

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

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A344328 Number of divisors of n^5.

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