A247516 Card{(x,y,z,t): 1<=x,y,z,t<=n, gcd(x,y,z,t)=1, lcm(x,y,z,t)=n}.

1, 14, 14, 50, 14, 196, 14, 110, 50, 196, 14, 700, 14, 196, 196, 194, 14, 700, 14, 700, 196, 196, 14, 1540, 50, 196, 110, 700, 14, 2744, 14, 302, 196, 196, 196, 2500, 14, 196, 196, 1540, 14, 2744, 14, 700, 700, 196, 14, 2716, 50, 700, 196, 700, 14, 1540, 196

Offset: 1

Author

Ovidiu Bagdasar, Sep 18 2014

Keywords

• nonn
• easy
• mult

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.

Crossrefs

Cf. A034444 (produced by L(n,2)), A245019, A070920.

Programs

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 *)

PARI

a(n) = {f = factor(n); 2^omega(n)*prod(k=1, #f~, 6*f[k, 2]^2+1);} \\ Michel Marcus, Sep 18 2014

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).

Extensions

More terms from Michel Marcus, Sep 18 2014