A061704 -id:A061704 - cp's OEIS frontend

A333843 Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4

Offset: 1

Ilya Gutkovskiy, Apr 07 2020

Sum of cube roots of cube divisors of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Atul Dixit, Bibekananda Maji, and Akshaa Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=3,k=3,n).

Cf. A000203, A053150, A061704, A069290, A113061, A333844.

nmax = 108; CoefficientList[Series[Sum[k x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/3) &, IntegerQ[#^(1/3)] &], {n, 108}]
f[p_, e_] := (p^(Floor[e/3] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

a(n) = {my(f = factor(n)); prod(i=1, #f~, (f[i,1]^(f[i,2]\3 + 1) - 1)/(f[i,1] - 1));} \\ Amiram Eldar, Sep 05 2023

Dirichlet g.f.: zeta(s) * zeta(3*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/3) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ Pi^2*n/6 + zeta(2/3)*n^(2/3)/2. - Vaclav Kotesovec, Dec 01 2020
a(n) = A000203(A053150(n)) (the sum of divisors of the cube root of largest cube dividing n). - Amiram Eldar, Sep 05 2023

A248780 Number of cubes that divide n!

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 8, 8, 8, 24, 36, 36, 36, 36, 42, 112, 112, 112, 128, 192, 192, 240, 270, 270, 270, 270, 330, 792, 792, 792, 864, 864, 864, 2016, 2912, 2912, 4704, 4704, 4704, 5376, 5760, 5760, 6144, 6144, 7680, 15360, 16320, 16320, 18360

Offset: 1

Clark Kimberling, Oct 15 2014

			a(9) counts these divisors of 9!:  1, 8, 27, 64, 216, 1728.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000142, A055993, A061704, A248781, A248762.

z = 130; m = 3; f[n_] := f[n] = FactorInteger[n!];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
a[n_] := Apply[Times, 1 + Floor[v[n]/m]]
A248780 = Table[a[n], {n, 1, z}] (* simplified by M. F. Hasler, Oct 22 2014 *)

a(n)=sumdiv(n!,d,ispower(d,3))
for(n=1,50,print1(a(n),", ")) \\ Derek Orr, Oct 20 2014, simplified by M. F. Hasler, Oct 22 2014

A248780(n)=prod(i=1,#n=factor(n!)[,2],1+n[i]\3) \\ M. F. Hasler, Oct 22 2014

a(n) = product_{i=1..r} 1+floor(e[i]/3), where product_{i=1..r} p[i]^e[i] is the prime factorization of n!. - M. F. Hasler, Oct 22 2014

a(n) = A061704(A000142(n)). - Michel Marcus, Mar 27 2015

A380395 The number of unitary divisors of n that are cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 23 2025

First differs from A318672 at n = 64.

The sum of unitary divisors of n that are cubes is A380396(n).

			a(8) = 2 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3.
a(216) = 4 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3.

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

Cf. A000578, A046100, A056624, A061704, A077610, A307427, A318672, A366761, A380396, A380397.

Cf. A002117, A013662.

f[p_, e_] := 2^If[Divisible[e, 3], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^if(f[i, 2]%3, 0, 1));}

a(n) = Sum_{d|n, gcd(d, n/d) = 1} [d is cube], where [] is the Iverson bracket.
Multiplicative with a(p^e) = 2 is e is divisible by 3, and 1 otherwise.
a(n) = abs(A307427(n)).
a(n) = A061704(n) - A380397(n).
a(n) >= 1, with equality if and only if n is not in A366761.
a(n) <= A061704(n), with equality if and only if n is biquadratefree (A046100).
Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(4*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(4) = 1.11062653532614811717... .
In general, the asymptotic mean of the number of unitary divisors of n that are m-th powers is zeta(m)/zeta(m+1), for m >= 2.

A309082 a(n) = n - floor(n/2^3) + floor(n/3^3) - floor(n/4^3) + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67

Offset: 1

Ilya Gutkovskiy, Jul 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A013937, A059851, A061704, A309081, A309083.

Table[Sum[(-1)^(k + 1) Floor[n/k^3], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
Table[Sum[Boole[IntegerQ[d^(1/3)] && OddQ[d]], {d, Divisors[n]}] - Sum[Boole[IntegerQ[d^(1/3)] && EvenQ[d]], {d, Divisors[n]}], {n, 1, 75}] // Accumulate

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

a(n) ~ 3*zeta(3)*n/4. - Vaclav Kotesovec, Oct 12 2019

A281613 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) / Product_{j>=1} (1 - x^(j^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 30, 33, 36, 39, 42, 45, 48, 54, 58, 62, 67, 72, 77, 82, 87, 96, 102, 108, 116, 123, 130, 137, 144, 156, 164, 172, 183, 192, 201, 210, 219, 234, 244, 254, 268, 279, 290, 303, 315, 334, 347, 360, 378, 392, 406, 423, 438, 462, 479, 496, 519, 537, 555, 577

Offset: 1

Ilya Gutkovskiy, Jan 25 2017

nonn

Total number of parts in all partitions of n into cubes.

Convolution of A003108 and A061704.

			a(9) = 11 because we have [8, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1] and 2 + 9 = 11.

Index entries for related partition-counting sequences

Cf. A000578, A003108, A061704, A281541.

nmax = 70; Rest[CoefficientList[Series[Sum[x^i^3/(1 - x^i^3), {i, 1, nmax}]/Product[1 - x^j^3, {j, 1, nmax}], {x, 0, nmax}], x]]

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

A295657 Multiplicative with a(p^e) = p^floor((e-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 28 2017

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000188, A003557, A004526, A020639, A028234, A067029, A295658, A295659.
Cf. A004709 (positions of ones), A082020, A212793.
Cf. also A008834, A053150, A061704.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> p^Floor[(e - 1)/2]] &, 105] (* Michael De Vlieger, Nov 28 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^floor((f[i,2]-1)/2));} \\ Amiram Eldar, Nov 30 2022

a(1) = 1; for n > 1, a(n) = A020639(n)^A004526(A067029(n)-1) * a(A028234(n)).
a(n) = A000188(A003557(n)).
a(n) = 1 iff A212793(n) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/Pi^2 = 1.519817... (A082020). - Amiram Eldar, Nov 30 2022

A291208 Number of noncube divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 6, 2, 3, 2, 5, 1, 7, 1, 4, 3, 3, 3, 8, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 8, 2, 5, 3, 5, 1, 6, 3, 6, 3, 3, 1, 11, 1, 3, 5, 4, 3, 7, 1, 5, 3, 7, 1, 10, 1, 3, 5, 5, 3, 7, 1, 8, 3, 3, 1, 11, 3, 3, 3, 6, 1, 11, 3, 5, 3, 3, 3, 10, 1, 5, 5, 8, 1, 7, 1, 6, 7

Offset: 1

Ilya Gutkovskiy, Aug 21 2017

nonn

			a(8) = 2 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are noncubes {2, 4}.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A007412, A056595, A061704.

Cf. A001620, A002117.

nmax = 105; Rest[CoefficientList[Series[Sum[(x^k - x^k^3)/((1 - x^k) (1 - x^k^3)), {k, 1, nmax}], {x, 0, nmax}], x]]
f1[p_, e_] := e + 1; f2[p_, e_] := 1 + Floor[e/3]; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f1 @@@ fct - Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

a(n) = sumdiv(n, d, !ispower(d, 3)); \\ Michel Marcus, Aug 21 2017

from math import prod
from sympy import factorint
def A291208(n):
    f = factorint(n).values()
    return prod(e+1 for e in f)-prod(e//3+1 for e in f) # Chai Wah Wu, Jun 05 2025

G.f.: Sum_{k>=1} x^A007412(k)/(1 - x^A007412(k)).
G.f.: Sum_{k>=1} (x^k - x^(k^3))/((1 - x^k)*(1 - x^(k^3))).
a(n) = A000005(n) - A061704(n).
From Amiram Eldar, Jan 30 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s) - zeta(3*s)).
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(3) - 1), where gamma is Euler's constant (A001620). (End)

A294333 Number of partitions of n into cubes dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 3, 1, 8, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 14, 1

Offset: 0

Ilya Gutkovskiy, Oct 28 2017

nonn

			a(8) = 2 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are cubes {1, 8} therefore we have [8] and [1, 1, 1, 1, 1, 1, 1, 1].

Cf. A018818, A003108, A004709, A030078, A061704, A284345.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] == 0 && IntegerQ[k^(1/3)]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 105}]

cubes_dividing(n) = select(d -> ispower(d,3),divisors(n));
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A294333(n) = if(n<2,1,partitions_into(n,vecsort(cubes_dividing(n), , 4))); \\ Antti Karttunen, Jul 21 2018

a(n) = 1 if n is a cubefree.

a(n) = 2 if n is a cube of prime.

A359944 Number of divisors d of n such that d-1 is a cube.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

The Cartesian equation for the Folium of Descartes is given as x^3 + y^3 = 3*k*x*y. If we set 3*k = n, then a(n)-1 is the number of integer solutions such that x,y > 0 and y >= x. Let d = m^3+1 be a divisor of n, then x = 3*k*m/(m^3+1); y = 3*k*m^2/(m^3+1) is a solution. - Thomas Scheuerle, Aug 07 2024

Cf. A001093, A061704, A339606.

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

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

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(k^3+1) = 1 + A339606 = 1.686503... . - Amiram Eldar, Jan 01 2024

A365487 The number of divisors of the largest cube dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 05 2023

The number of divisors of the cube root of the largest cube dividing n, A053150(n), is A061704(n).

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

Cf. A000005, A000578, A004709, A008834, A053150, A061704.

f[p_, e_] := 3*Floor[e/3] + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = vecprod(apply(x -> 3*(x\3) + 1, factor(n)[, 2]));

a(n) = A000005(A008834(n)).
Multiplicative with a(p^e) = 3*floor(e/3) + 1.
a(n) = 1 if and only if n is cubefree (A004709).
a(n) <= A000005(n) with equality if and only if n is a cube (A000578).
Dirichlet g.f.: zeta(s) * zeta(3*s) * Product_{p prime} (1 + 2/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3) * Product_{p prime} (1 + 2/p^3) = 1.6552343865608... .

cp's OEIS Frontend

A333843 Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).

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A248780 Number of cubes that divide n!

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A380395 The number of unitary divisors of n that are cubes.

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A309082 a(n) = n - floor(n/2^3) + floor(n/3^3) - floor(n/4^3) + ...

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A281613 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) / Product_{j>=1} (1 - x^(j^3)).

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A295657 Multiplicative with a(p^e) = p^floor((e-1)/2).

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A291208 Number of noncube divisors of n.

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A294333 Number of partitions of n into cubes dividing n.

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