A247513 Number of elements in the set {(x,y,z): 1<=x,y,z<=n, gcd(x,y,z)=1, lcm(x,y,z)=n}.

1, 6, 6, 12, 6, 36, 6, 18, 12, 36, 6, 72, 6, 36, 36, 24, 6, 72, 6, 72, 36, 36, 6, 108, 12, 36, 18, 72, 6, 216, 6, 30, 36, 36, 36, 144, 6, 36, 36, 108, 6, 216, 6, 72, 72, 36, 6, 144, 12, 72, 36, 72, 6, 108, 36, 108, 36, 36, 6, 432

Offset: 1

Author

Ovidiu Bagdasar, Sep 18 2014

Keywords

• nonn
• mult
• easy

Comments

For given n and k positive integers, let L(n,k) represent the number of ordered k-tuples of positive integers whose GCD is 1 and LCM is n. In this notation, the sequence corresponds to a(n) = L(n,3).

The inverse Mobius transform is apparently in A070919. - R. J. Mathar, May 25 2017

Examples

			The triples corresponding to a(2)=6 are (1,1,2), (1,2,1), (2,1,1), (1,2,2), (2,1,2) and (2,2,1).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• O. Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific publications of the state university of Novi Pazar, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, No. 2 (2014), pp. 91-100.

Crossrefs

L(n,2) produces A034444.

Programs

Maple

a:= proc(n) local F; F:= ifactors(n)[2];
      mul(6*f[2],f=F)
end proc:
seq(a(n),n=1..40); # Robert Israel, Sep 22 2014

Mathematica

a[n_] := 6^PrimeNu[n] Times @@ FactorInteger[n][[All, 2]];
Array[a, 60] (* Jean-François Alcover, Jul 27 2020 *)
a[1] = 1; a[n_] := Times @@ (6 * Last[#]& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

PARI

a(n) = {f = factor(n); 6^omega(n)*prod(k=1, #f~, f[k, 2]); }

Formula

For n = p_1^{n_1} p_2^{n_2}...p_r^{n_r} one has

a(n) = Product_{i=1..r} ((n_i+1)^3 - 2*n_i^3 + (n_i-1)^3).

a(n) = 6^omega(n)*Product_{i=1..r} n_i.

a(n) = 6^A001221(n) *A005361(n). - R. J. Mathar, May 25 2017

Multiplicative with a(p^e) = 6*e. - Amiram Eldar, Sep 26 2020