A086222 a(n) = card{ (x,y,z) | x <= y <= z and lcm(x,y,z) = n }.

1, 3, 3, 6, 3, 13, 3, 10, 6, 13, 3, 30, 3, 13, 13, 15, 3, 30, 3, 30, 13, 13, 3, 54, 6, 13, 10, 30, 3, 71, 3, 21, 13, 13, 13, 73, 3, 13, 13, 54, 3, 71, 3, 30, 30, 13, 3, 85, 6, 30, 13, 30, 3, 54, 13, 54, 13, 13, 3, 178, 3, 13, 30, 28, 13, 71, 3, 30, 13, 71, 3, 135, 3, 13, 30, 30, 13, 71, 3

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

Keywords

• nonn
• easy

Comments

A number of conjectures are possible, many of which should be easy to prove. Examples: (1) If n is a product of two primes then a(n)=13. (2) If n is a square of a prime then a(n)=6. - John W. Layman, Sep 01 2003

Links

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

Crossrefs

Cf. A048691, A070919, A018892, A086165, A377304, A008683.

Programs

Mathematica

f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 1; 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));
A086222(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

For a prime p, a(p) = 3.

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

a(n) = Sum_{d|n} A377304(d)*A008683(n/d). - Ridouane Oudra, May 22 2025

a(n) = A086165(n) + A048691(n). - Ridouane Oudra, Aug 19 2025

Extensions

More terms from John W. Layman, Sep 01 2003