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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}.

%I A247513 #43 Sep 26 2020 08:00:49
%S A247513 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 A247513 18,72,6,216,6,30,36,36,36,144,6,36,36,108,6,216,6,72,72,36,6,144,12,
%U A247513 72,36,72,6,108,36,108,36,36,6,432
%N 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}.
%C A247513 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 A247513 The inverse Mobius transform is apparently in A070919. - _R. J. Mathar_, May 25 2017
%H A247513 Antti Karttunen, <a href="/A247513/b247513.txt">Table of n, a(n) for n = 1..10000</a>
%H A247513 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 A247513 For n = p_1^{n_1} p_2^{n_2}...p_r^{n_r} one has
%F A247513 a(n) = Product_{i=1..r} ((n_i+1)^3 - 2*n_i^3 + (n_i-1)^3).
%F A247513 a(n) = 6^omega(n)*Product_{i=1..r} n_i.
%F A247513 a(n) = 6^A001221(n) *A005361(n). - _R. J. Mathar_, May 25 2017
%F A247513 Multiplicative with a(p^e) = 6*e. - _Amiram Eldar_, Sep 26 2020
%e A247513 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 A247513 a:= proc(n) local F; F:= ifactors(n)[2];
%p A247513       mul(6*f[2],f=F)
%p A247513 end proc:
%p A247513 seq(a(n),n=1..40); # _Robert Israel_, Sep 22 2014
%t A247513 a[n_] := 6^PrimeNu[n] Times @@ FactorInteger[n][[All, 2]];
%t A247513 Array[a, 60] (* _Jean-François Alcover_, Jul 27 2020 *)
%t A247513 a[1] = 1; a[n_] := Times @@ (6 * Last[#]& /@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 26 2020 *)
%o A247513 (PARI) a(n) = {f = factor(n); 6^omega(n)*prod(k=1, #f~, f[k, 2]); }
%Y A247513 L(n,2) produces A034444.
%K A247513 nonn,mult,easy
%O A247513 1,2
%A A247513 _Ovidiu Bagdasar_, Sep 18 2014