A113061 -id:A113061 - cp's OEIS frontend

A370240 The sum of divisors of n that are cubes of squarefree numbers.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 28, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 13 2024

First differs from A366904 at n = 32, and from A113061 at n = 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A062838, A113061, A366904, A370239.

f[p_, e_] := If[e <= 2, 1, 1 + p^3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= 2, 1, 1 + f[i,1]^3));}

Multiplicative with a(p^e) = 1 for e <= 2, and a(p^e) = 1 + p^3 for e >= 3.
Dirichlet g.f.: zeta(s)*zeta(3*s-3)/zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(4/3) + n, where c = 3*zeta(4/3)/(2*Pi^2) = 0.5472769126... .

A342229 Total sum of parts which are cubes in all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 19, 30, 53, 75, 113, 163, 235, 328, 461, 628, 868, 1163, 1564, 2069, 2743, 3578, 4674, 6036, 7795, 9962, 12728, 16151, 20441, 25714, 32290, 40332, 50292, 62405, 77288, 95339, 117382, 143987, 176298, 215168, 262121, 318385, 386043, 466838, 563577, 678712

Offset: 0

Ilya Gutkovskiy, Mar 06 2021

nonn

			For n = 4 we have:
--------------------------------
Partitions        Sum of parts
.               which are cubes
--------------------------------
4 ................... 0
3 + 1 ............... 1
2 + 2 ............... 0
2 + 1 + 1 ........... 2
1 + 1 + 1 + 1 ....... 4
--------------------------------
Total ............... 7
So a(4) = 7.

Cf. A000041, A000578, A066186, A113061, A264392, A342228.

nmax = 45; CoefficientList[Series[Sum[k^3 x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/3)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]

G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 - x^(k^3)) / Product_{j>=1} (1 - x^j).

a(n) = Sum_{k=1..n} A113061(k) * A000041(n-k).

A359943 a(n) = Sum_{d|n, d-1 is cube} d.

Original entry on oeis.org

1, 3, 1, 3, 1, 3, 1, 3, 10, 3, 1, 3, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 3, 1, 3, 10, 31, 1, 3, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 3, 1, 3, 10, 3, 1, 3, 1, 3, 1, 3, 1, 12, 1, 31, 1, 3, 1, 3, 1, 3, 10, 3, 66, 3, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 3, 1, 3, 10, 3, 1, 31, 1, 3

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

Cf. A001093, A113061, A359937, A359942, A359944.

a[n_] := DivisorSum[n, # &, IntegerQ[Surd[#-1, 3]] &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

PARI
```
a(n) = sumdiv(n, d, ispower(d-1, 3)*d);
```

G.f.: Sum_{k>=0} (k^3+1) * x^(k^3+1)/(1 - x^(k^3+1)).

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A370240 The sum of divisors of n that are cubes of squarefree numbers.

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A342229 Total sum of parts which are cubes in all partitions of n.

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A359943 a(n) = Sum_{d|n, d-1 is cube} d.

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