OFFSET
1,2
COMMENTS
For given n and k positive integers, let L(n,k) represent the number of ordered k-tuples of positive integers, whose GCD is 1 and LCM is n. In this notation, the sequence corresponds to a(n) = L(n,4).
LINKS
Alois P. Heinz, 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, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.
FORMULA
For n = p_1^{n_1} p_2^{n_2}...p_r^{n_r} one has
a(n) = Product_{i=1..r} ((n_i+1)^4 - 2*n_i^4 + (n_i-1)^4).
a(n) = 2^omega(n)*Product_{i=1..r} (6n_i^2 + 1).
MATHEMATICA
f[p_, e_] := 2*(6*e^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)
PROG
(PARI) a(n) = {f = factor(n); 2^omega(n)*prod(k=1, #f~, 6*f[k, 2]^2+1); } \\ Michel Marcus, Sep 18 2014
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Ovidiu Bagdasar, Sep 18 2014
EXTENSIONS
More terms from Michel Marcus, Sep 18 2014
STATUS
approved