A226602 Number of ordered triples (i,j,k) with i*j*k = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

1, 1, 3, 3, 6, 3, 9, 3, 9, 6, 9, 3, 18, 3, 9, 9, 12, 3, 18, 3, 18, 9, 9, 3, 27, 6, 9, 9, 18, 3, 27, 3, 15, 9, 9, 9, 36, 3, 9, 9, 27, 3, 27, 3, 18, 18, 9, 3, 36, 6, 18, 9, 18, 3, 27, 9, 27, 9, 9, 3, 54, 3, 9, 18, 18, 9, 27, 3, 18, 9, 27, 3, 54, 3, 9, 18, 18

Offset: 0

Author

Robert Price, Jun 13 2013

Keywords

• nonn
• mult

Comments

Note that gcd(0,m) = m for any m.

a(n) is the number of cubefree divisors summed over the divisors of n. In other words, a(n) = Sum_{d|n} A073184(d). - Geoffrey Critzer, Mar 20 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 101 terms from Robert Price)

Crossrefs

Cf. A000005, A007427, A008836, A034444, A056624, A073184, A092520, A100450, A212793, A226357, A226359.

Programs

Maple

with(numtheory):
b:= proc(n, t, g) option remember; `if`(t=0,
      `if`(igcd(n, g)=1, 1, 0), add(b(n/d, t-1,
      igcd(g, d)), d=divisors(n)))
    end:
a:= n-> `if`(n=0, 1, b(n, 2, 0)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2015

Mathematica

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n && GCD[i, j, k] <= 1, {i, j, k}], {i, 0,n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]
a[0] = a[1] = 1; a[n_] := Times @@ (3 * Last[#] & /@ FactorInteger[n]); Array[a, 100, 0] (* Amiram Eldar, Sep 14 2020 *)

Python

from math import prod
from sympy import factorint
def A226602(n): return prod(3*e for e in factorint(n).values()) if n else 1 # Chai Wah Wu, Dec 26 2022

Formula

From Geoffrey Critzer, Mar 20 2015: (Start)

If n = p_1^e_1*p_2^e_2*...*p_r^e_r then a(n) = Product_{i=1..r} 3*e_i.

Dirichlet g.f.: zeta(s)^3/zeta(3*s). (End)

From Werner Schulte, May 13 2018: (Start)

Multiplicative with a(p^e) = 3*e, p prime and e>0.

Dirichlet inverse b(n), n>0, is multiplicative with b(1) = 1, and for p prime and e>0: b(p^e)=0 if e mod 3 = 0 otherwise b(p^e)=3*(-1)^(e mod 3).

Dirichlet convolution with A007427(n) yields A212793(n).

Dirichlet convolution with A008836(n) yields A092520(n).

Equals Dirichlet convolution of A034444(n) and A056624(n).

Equals Dirichlet convolution of A000005(n) and A212793(n). (End)

Sum_{k=1..n} a(k) ~ n/(2*Zeta(3)) * (log(n)^2 + 2*log(n) * (-1 + 3*gamma - 3*Zeta'(3)/Zeta(3)) + 2 + 6*gamma^2 - 6*sg1 + 6*Zeta'(3)/Zeta(3) + 18*Zeta'(3)^2/Zeta(3)^2 - 6*gamma*(1 + 3*Zeta'(3)/Zeta(3)) - 9*Zeta''(3)/Zeta(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 07 2019

a(n) = A005361(n) * A074816(n). - Vaclav Kotesovec, Feb 27 2023