A247517 Card{(x,y,z,t,u): 1<=x,y,z,t,u<=n, gcd(x,y,z,t,u)=1, lcm(x,y,z,t,u)=n}.

1, 30, 30, 180, 30, 900, 30, 570, 180, 900, 30, 5400, 30, 900, 900, 1320, 30, 5400, 30, 5400, 900, 900, 30, 17100, 180, 900, 570, 5400, 30, 27000, 30, 2550, 900, 900, 900, 32400, 30, 900, 900, 17100, 30, 27000, 30, 5400, 5400, 900, 30, 39600, 180, 5400, 900

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,5).

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, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.

Crossrefs

L(n,2) produces A034444, A245019, A070921.

Cf. A247516.

Programs

Mathematica

f[p_, e_] := 10*(2*e^3 + e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

PARI

a(n) = {f = factor(n); 10^omega(n)*prod(k=1, #f~, 2*f[k, 2]^3+f[k, 2]); }

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)^5 - 2*n_i^5 + (n_i-1)^5).

a(n) = 10^omega(n)*Product_{i=1..r} (2n_i^3 + n_i).