A070919 a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.

1, 7, 7, 19, 7, 49, 7, 37, 19, 49, 7, 133, 7, 49, 49, 61, 7, 133, 7, 133, 49, 49, 7, 259, 19, 49, 37, 133, 7, 343, 7, 91, 49, 49, 49, 361, 7, 49, 49, 259, 7, 343, 7, 133, 133, 49, 7, 427, 19, 133, 49, 133, 7, 259, 49, 259, 49, 49, 7, 931, 7, 49, 133, 127, 49, 343, 7, 133

Offset: 1

Author

Benoit Cloitre, May 20 2002

Keywords

• mult
• easy
• nonn

Comments

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.

Crossrefs

Cf. A000005, A008683, A048691, A070920, A070921, A086222, A247513.

Programs

Mathematica

Join[{1},Table[Product[(k + 1)^3 - k^3, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

PARI

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^3*moebius(n/d)),","))

PARI

a(n) = vecprod(apply(x->(x+1)^3-x^3, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

Formula

a(n) = Sum_{d|n} A000005(d)^3*A008683(n/d).

Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^3/k^s.

Multiplicative with a(p^e) = 1+3*e+3*e^2 for prime p and e >= 0. - Werner Schulte, Nov 30 2018