A070920 a(n) = Card{ (x,y,z,u) | lcm(x,y,z,u)=n }.

1, 15, 15, 65, 15, 225, 15, 175, 65, 225, 15, 975, 15, 225, 225, 369, 15, 975, 15, 975, 225, 225, 15, 2625, 65, 225, 175, 975, 15, 3375, 15, 671, 225, 225, 225, 4225, 15, 225, 225, 2625, 15, 3375, 15, 975, 975, 225, 15, 5535, 65, 975, 225, 975, 15, 2625, 225

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, A070919, A070921, A247516 (Mobius transform).

Programs

Mathematica

Join[{1},Table[Product[(k + 1)^4 - k^4, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

PARI

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^4*moebius(n/d)),","))

PARI

a(n) = vecprod(apply(x->(x+1)^4-x^4, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

Formula

a(n) = Sum_{d|n} A000005(d)^4*A008683(n/d).

Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^4/k^s.

Multiplicative with a(p^e) = (e+1)^4 - e^4. - Amiram Eldar, Sep 03 2023