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A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

%I A086165 #26 Aug 18 2025 22:25:27
%S A086165 0,0,0,1,0,4,0,3,1,4,0,15,0,4,4,6,0,15,0,15,4,4,0,33,1,4,3,15,0,44,0,
%T A086165 10,4,4,4,48,0,4,4,33,0,44,0,15,15,4,0,58,1,15,4,15,0,33,4,33,4,4,0,
%U A086165 133,0,4,15,15,4,44,0,15,4,44,0,100,0,4,15,15,4,44,0,58,6,4,0,133,4,4,4,33,0
%N A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.
%H A086165 Antti Karttunen, <a href="/A086165/b086165.txt">Table of n, a(n) for n = 1..10000</a>
%F A086165 a(n) = (A070919(n) - 3*A048691(n) + 2)/6. - _Vladeta Jovovic_, Dec 01 2004
%F A086165 a(n) = A086222(n) - A048691(n). - _Ridouane Oudra_, Aug 14 2025
%p A086165 for n from 1 to 100 do a[n] := 0:for x from 1 to n do for y from x+1 to n do for z from y+1 to n do if(lcm(x,y,z)=n) then a[n] := a[n]+1:fi:od:od:od:od:seq(a[j],j=1..200); # _Sascha Kurz_, Sep 22 2003
%t A086165 f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) - 3 * Times @@ f2 @@@f + 2) / 6; Array[a, 100] (* _Amiram Eldar_, Sep 03 2023 *)
%o A086165 (PARI)
%o A086165 A048691(n) = numdiv(n^2);
%o A086165 A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));
%o A086165 A086165(n) = ((A070919(n)-3*A048691(n)+2)/6); \\ _Antti Karttunen_, May 19 2017, after Jovovic's formula
%o A086165 (PARI) a(n) = {my(e = factor(n)[, 2]); (vecprod(apply(x->(x+1)^3-x^3, e)) - 3*vecprod(apply(x->2*x+1, e)) + 2) / 6;} \\ _Amiram Eldar_, Sep 03 2023
%Y A086165 Cf. A048691, A070919, A086222.
%K A086165 nonn,easy
%O A086165 1,6
%A A086165 Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003
%E A086165 More terms from _Sascha Kurz_, Sep 22 2003