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

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, 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, 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

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048691, A070919, A086222.

Programs

Maple

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

Mathematica

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 *)

PARI

A048691(n) = numdiv(n^2);
A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));
A086165(n) = ((A070919(n)-3*A048691(n)+2)/6); \\ Antti Karttunen, May 19 2017, after Jovovic's formula

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

Formula

a(n) = (A070919(n) - 3*A048691(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004

a(n) = A086222(n) - A048691(n). - Ridouane Oudra, Aug 14 2025

Extensions

More terms from Sascha Kurz, Sep 22 2003