A333843 -id:A333843 - cp's OEIS frontend

A333844 G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).

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

Offset: 1

Ilya Gutkovskiy, Apr 07 2020

Sum of 4th roots of 4th powers dividing n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=1,k=4,n).

Cf. A000203, A053164, A063775, A069290, A300909, A333843.

nmax = 112; CoefficientList[Series[Sum[k x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/4) &, IntegerQ[#^(1/4)] &], {n, 112}]
f[p_, e_] := (p^(Floor[e/4] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

Dirichlet g.f.: zeta(s) * zeta(4*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/4) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ zeta(3)*n + zeta(1/2)*sqrt(n)/2. - Vaclav Kotesovec, Dec 01 2020

A345321 Sum of the divisors of n whose cube does not divide n.

Original entry on oeis.org

0, 2, 3, 6, 5, 11, 7, 12, 12, 17, 11, 27, 13, 23, 23, 28, 17, 38, 19, 41, 31, 35, 23, 57, 30, 41, 36, 55, 29, 71, 31, 60, 47, 53, 47, 90, 37, 59, 55, 87, 41, 95, 43, 83, 77, 71, 47, 121, 56, 92, 71, 97, 53, 116, 71, 117, 79, 89, 59, 167, 61, 95, 103, 120, 83, 143, 67, 125, 95

Offset: 1

Wesley Ivan Hurt, Jun 13 2021

nonn

			a(16) = 28; The divisors of 16 whose cube does not divide 16 are: 4, 8 and 16. The sum of these divisors is then 4 + 8 + 16 = 28.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A333843.

Table[Sum[k (Ceiling[n/k^3] - Floor[n/k^3]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
Table[Total[Select[Divisors[n],Mod[n,#^3]!=0&]],{n,100}] (* Harvey P. Dale, May 01 2022 *)

a(n) = sumdiv(n, d, if (n % d^3, d)); \\ Michel Marcus, Jun 13 2021
(Python 3.8+)
from math import prod
from sympy import factorint
def A345321(n):
    f = factorint(n).items()
    return prod((p**(q+1)-1)//(p-1) for p, q in f) - prod((p**(q//3+1)-1)//(p-1) for p, q in f) # Chai Wah Wu, Jun 14 2021

a(n) = Sum_{k=1..n} k * (ceiling(n/k^3) - floor(n/k^3)) * (1 - ceiling(n/k) + floor(n/k)).

a(n) = A000203(n) - A333843(n). - Rémy Sigrist, Jun 14 2021

A361793 Sum of the squares d^2 of the divisors d satisfying d^3|n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 10, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 1

R. J. Mathar, Mar 24 2023

The Mobius transform is 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, ... = n^(2/3)*A010057(n).

Winston de Greef, Table of n, a(n) for n = 1..10000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=2,k=3,n).

Cf. A035316, A069290, A333843.

gsigma := proc(n,z,k)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(n,d^k) = 0 then
            a := a+d^z ;
        end if ;
    end do:
    a ;
end proc:
seq( gsigma(n,2,3),n=1..80) ;

f[p_, e_] := (p^(2*(Floor[e/3] + 1)) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 24 2023 *)

a(n) = sumdiv(n, d, if (ispower(d, 3), sqrtnint(d, 3)^2)); \\ Michel Marcus, Mar 24 2023

for(n=1, 100, print1(direuler(p=2, n, (1/((1-X)*(1-p^2*X^3))))[n], ", ")) \\ Vaclav Kotesovec, Jun 26 2024

from math import prod
from sympy import factorint
def A361793(n): return prod((p**(e//3+1<<1)-1)//(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023

a(n) = Sum_{d^3|n} d^2.
Multiplicative with a(p^e) = (p^(2*(floor(e/3) + 1)) - 1)/(p^2 - 1). - Amiram Eldar, Mar 24 2023
G.f.: Sum_{k>=1} k^2 * x^(k^3) / (1 - x^(k^3)). - Ilya Gutkovskiy, Jun 05 2024
From Vaclav Kotesovec, Jun 26 2024: (Start)
Dirichlet g.f.: zeta(s)*zeta(3*s-2).
Sum_{k=1..n} a(k) ~ n*(log(n) + 4*gamma - 1)/3, where gamma is the Euler-Mascheroni constant A001620. (End)

A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 9, 28, 1, 1, 9, 1, 1, 1, 73, 1, 28, 1, 9, 1, 1, 1, 9, 126, 1, 28, 9, 1, 1, 1, 73, 1, 1, 1, 252, 1, 1, 1, 9, 1, 1, 1, 9, 28, 1, 1, 73, 344, 126, 1, 9, 1, 28, 1, 9, 1, 1, 1, 9, 1, 1, 28, 585, 1, 1, 1, 9, 1, 1, 1, 252, 1, 1, 126, 9, 1, 1, 1, 73

Offset: 1

R. J. Mathar, Mar 24 2023

The Mobius transform is 1, 0, 0, 8, 0, 0, 0, 0, 27, 0, 0, ... = n^(3/2)*A010052(n).

Winston de Greef, Table of n, a(n) for n = 1..10000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=3,k=2,n).

Cf. A010052, A035316, A069290, A351308, A333843, A333844.

gsigma := proc(n,z,k)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(n,d^k) = 0 then
            a := a+d^z ;
        end if ;
    end do:
    a ;
end proc:
seq( gsigma(n,3,2),n=1..80) ;

f[p_, e_] := (p^(3*(Floor[e/2] + 1)) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 24 2023 *)

a(n) = sumdiv(n, d, if (issquare(d), sqrtint(d)^3)); \\ Michel Marcus, Mar 24 2023

from math import prod
from sympy import factorint
def A361794(n): return prod((p**(3*(e>>1)+3)-1)//(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023

a(n) = Sum_{d^2|n} d^3.
Multiplicative with a(p^e) = (p^(3*(floor(e/2) + 1)) - 1)/(p^3 - 1). - Amiram Eldar, Mar 24 2023
G.f.: Sum_{k>=1} k^3 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024

A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 30 2023

First differs from A295659 at n = 64.

The number of distinct prime factors of the cube root of the largest cube dividing n is A295659(n).

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

Cf. A000578, A001222, A002264, A008834, A004709, A053150, A286229, A295659.

Cf. A061704 (number of divisors), A333843 (sum of divisors).

f[p_, e_] := Floor[e/3]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> x\3, factor(n)[, 2]));

a(n) = A001222(A053150(n)).
a(n) = A001222(A008834(n))/3.
Additive with a(p^e) = floor(e/3) = A002264(e).
a(n) >= 0, with equality if and only if n is cubefree (A004709).
a(n) <= A001222(n)/3, with equality if and only if n is a positive cube (A000578 \ {0}).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^3-1) = 0.194118... (A286229).

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A333844 G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).

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A345321 Sum of the divisors of n whose cube does not divide n.

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A361793 Sum of the squares d^2 of the divisors d satisfying d^3|n.

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A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.

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A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.

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