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Number of elements in the set {(x,y,z): 1<=x,y,z<=n, gcd(x,y,z)=1, lcm(x,y,z)=n}.
2

%I #43 Sep 26 2020 08:00:49

%S 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,

%T 18,72,6,216,6,30,36,36,36,144,6,36,36,108,6,216,6,72,72,36,6,144,12,

%U 72,36,72,6,108,36,108,36,36,6,432

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

%C 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).

%C The inverse Mobius transform is apparently in A070919. - _R. J. Mathar_, May 25 2017

%H Antti Karttunen, <a href="/A247513/b247513.txt">Table of n, a(n) for n = 1..10000</a>

%H O. Bagdasar, <a href="http://hdl.handle.net/10545/583372">On Some Functions Involving the lcm and gcd of Integer Tuples</a>, Scientific publications of the state university of Novi Pazar, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, No. 2 (2014), pp. 91-100.

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

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

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

%F a(n) = 6^A001221(n) *A005361(n). - _R. J. Mathar_, May 25 2017

%F Multiplicative with a(p^e) = 6*e. - _Amiram Eldar_, Sep 26 2020

%e 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).

%p a:= proc(n) local F; F:= ifactors(n)[2];

%p mul(6*f[2],f=F)

%p end proc:

%p seq(a(n),n=1..40); # _Robert Israel_, Sep 22 2014

%t a[n_] := 6^PrimeNu[n] Times @@ FactorInteger[n][[All, 2]];

%t Array[a, 60] (* _Jean-François Alcover_, Jul 27 2020 *)

%t a[1] = 1; a[n_] := Times @@ (6 * Last[#]& /@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 26 2020 *)

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

%Y L(n,2) produces A034444.

%K nonn,mult,easy

%O 1,2

%A _Ovidiu Bagdasar_, Sep 18 2014