A048785 -id:A048785 - cp's OEIS frontend

A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

1, 5, 5, 10, 5, 25, 5, 15, 10, 25, 5, 50, 5, 25, 25, 20, 5, 50, 5, 50, 25, 25, 5, 75, 10, 25, 15, 50, 5, 125, 5, 25, 25, 25, 25, 100, 5, 25, 25, 75, 5, 125, 5, 50, 50, 25, 5, 100, 10, 50, 25, 50, 5, 75, 25, 75, 25, 25, 5, 250, 5, 25, 50, 30, 25, 125, 5, 50, 25, 125, 5, 150

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e).

Cf. A082476.

g[p_, e_] := 5*e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+3*X+X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + 5*p^s/(p^s - 1)^2).

a(n) = A005361(n) * A082476(n).

A353551 a(n) = Sum_{k=1..n} tau(k^3), where tau is the number of divisors function A000005.

Original entry on oeis.org

0, 1, 5, 9, 16, 20, 36, 40, 50, 57, 73, 77, 105, 109, 125, 141, 154, 158, 186, 190, 218, 234, 250, 254, 294, 301, 317, 327, 355, 359, 423, 427, 443, 459, 475, 491, 540, 544, 560, 576, 616, 620, 684, 688, 716, 744, 760, 764, 816, 823, 851, 867, 895, 899, 939, 955, 995

Offset: 0

Karl-Heinz Hofmann, May 07 2022

nonn

			  A048785(0) = 0
+ A048785(1) = 1
+ A048785(2) = 4
+ A048785(3) = 4
------------------
= A353551(3) = 9

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Partial sums of A048785.

Cf. A000005, A006218, A061503 (squares).

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+numtheory[tau](n^3)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 08 2022

Accumulate[Join[{0}, Table[DivisorSigma[0, k^3], {k, 1, 50}]]] (* Amiram Eldar, May 08 2022 *)

a(n) = sum(k=1, n, numdiv(k^3)); \\ Michel Marcus, May 08 2022

from sympy import divisor_count
def A048785(n): return divisor_count(n**3)
def A353551(n): return sum(A048785(n) for n in range(1, n))
print([A353551(n) for n in range(1, 58)])

from math import prod
from sympy import factorint
def A353551(n): return sum(prod(3*e+1 for e in factorint(k).values()) for k in range(1,n+1)) # Chai Wah Wu, May 10 2022

a(n) = Sum_{k=1..n} tau(k^3).

a(n) = a(n-1) + A048785(n) for n >= 1, a(0) = 0.

A294072 Number of noncube divisors of n^3.

Original entry on oeis.org

0, 2, 2, 4, 2, 12, 2, 6, 4, 12, 2, 22, 2, 12, 12, 8, 2, 22, 2, 22, 12, 12, 2, 32, 4, 12, 6, 22, 2, 56, 2, 10, 12, 12, 12, 40, 2, 12, 12, 32, 2, 56, 2, 22, 22, 12, 2, 42, 4, 22, 12, 22, 2, 32, 12, 32, 12, 12, 2, 100, 2, 12, 22, 12, 12, 56, 2, 22, 12, 56, 2, 58, 2, 12, 22, 22, 12, 56, 2, 42, 8, 12, 2, 100, 12

Offset: 1

Ilya Gutkovskiy, Feb 07 2018

nonn

All terms are even. a(n)=2 if and only if n is prime. - Robert Israel, Jan 16 2020

			a(4) = 4 because 4^3 = 64 has 7 divisors {1, 2, 4, 8, 16, 32, 64} among which 4 divisors {2, 4, 16, 32} are noncubes.

Cf. A000005, A000578, A001221, A007412, A048785, A055205, A061704, A291208.

f:= proc(n) local F;
  F:= map(t -> t[2],ifactors(n)[2]);
  mul(1+3*t,t=F) - mul(1+t,t=F)
end proc:
map(f, [$1..100]; # Robert Israel, Jan 16 2020

nmax = 85; Rest[CoefficientList[Series[Sum[(3^PrimeNu[k] - 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Length[Select[Divisors[n], ! IntegerQ[#^(1/3)] &]]; Table[a[n^3], {n, 1, 85}]
Table[DivisorSigma[0, n^3] - DivisorSigma[0, n], {n, 1, 85}]

G.f.: Sum_{k>=1} (3^omega(k) - 1)*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221).
a(n) = [x^(n^3)] Sum_{k>=1} x^A007412(k)/(1 - x^A007412(k)).
a(n) = A291208(A000578(n)).
a(n) = A048785(n) - A000005(n).

A325339 Number of divisors of n^3 that are <= n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 6, 2, 8, 2, 6, 5, 5, 2, 9, 2, 8, 5, 6, 2, 11, 3, 6, 4, 8, 2, 17, 2, 6, 6, 6, 5, 14, 2, 6, 6, 11, 2, 17, 2, 9, 8, 6, 2, 15, 3, 10, 6, 9, 2, 13, 5, 11, 6, 6, 2, 26, 2, 6, 8, 7, 5, 18, 2, 10, 6, 17, 2, 18, 2, 6, 9, 10, 5, 19, 2, 14, 5

Offset: 1

Clark Kimberling, Apr 21 2019

			a(10) counts these 6 divisors: 1,2,4,5,8,10.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000578, A048785.

Table[Length[Select[Divisors[n^3], # <= n &]], {n, 1, 120}]

a(n) = sumdiv(n^3, d, d <= n); \\ Michel Marcus, Apr 22 2019

cp's OEIS Frontend

A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

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A353551 a(n) = Sum_{k=1..n} tau(k^3), where tau is the number of divisors function A000005.

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A294072 Number of noncube divisors of n^3.

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Maple

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A325339 Number of divisors of n^3 that are <= n.

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Mathematica

PARI