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

%I A226602 #44 Jun 13 2025 21:51:36
%S A226602 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,
%T A226602 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,
%U A226602 54,3,9,18,18,9,27,3,18,9,27,3,54,3,9,18,18
%N A226602 Number of ordered triples (i,j,k) with i*j*k = n, i,j,k >= 0 and gcd(i,j,k) <= 1.
%C A226602 Note that gcd(0,m) = m for any m.
%C A226602 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
%H A226602 Alois P. Heinz, <a href="/A226602/b226602.txt">Table of n, a(n) for n = 0..20000</a> (first 101 terms from Robert Price)
%F A226602 From _Geoffrey Critzer_, Mar 20 2015: (Start)
%F A226602 If n = p_1^e_1*p_2^e_2*...*p_r^e_r then a(n) = Product_{i=1..r} 3*e_i.
%F A226602 Dirichlet g.f.: zeta(s)^3/zeta(3*s). (End)
%F A226602 From _Werner Schulte_, May 13 2018: (Start)
%F A226602 Multiplicative with a(p^e) = 3*e, p prime and e>0.
%F A226602 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).
%F A226602 Dirichlet convolution with A007427(n) yields A212793(n).
%F A226602 Dirichlet convolution with A008836(n) yields A092520(n).
%F A226602 Equals Dirichlet convolution of A034444(n) and A056624(n).
%F A226602 Equals Dirichlet convolution of A000005(n) and A212793(n). (End)
%F A226602 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
%F A226602 a(n) = A005361(n) * A074816(n). - _Vaclav Kotesovec_, Feb 27 2023
%p A226602 with(numtheory):
%p A226602 b:= proc(n, t, g) option remember; `if`(t=0,
%p A226602       `if`(igcd(n, g)=1, 1, 0), add(b(n/d, t-1,
%p A226602       igcd(g, d)), d=divisors(n)))
%p A226602     end:
%p A226602 a:= n-> `if`(n=0, 1, b(n, 2, 0)):
%p A226602 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 20 2015
%t A226602 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}]
%t A226602 a[0] = a[1] = 1; a[n_] := Times @@ (3 * Last[#] & /@ FactorInteger[n]); Array[a, 100, 0] (* _Amiram Eldar_, Sep 14 2020 *)
%o A226602 (Python)
%o A226602 from math import prod
%o A226602 from sympy import factorint
%o A226602 def A226602(n): return prod(3*e for e in factorint(n).values()) if n else 1 # _Chai Wah Wu_, Dec 26 2022
%Y A226602 Cf. A000005, A007427, A008836, A034444, A056624, A073184, A092520, A100450, A212793, A226357, A226359.
%K A226602 nonn,mult
%O A226602 0,3
%A A226602 _Robert Price_, Jun 13 2013