A338798 a(n) = Sum_{k=1..n-1} lcm(lcm(n, k), lcm(n, n-k)).

0, 2, 12, 28, 100, 90, 392, 408, 792, 810, 2420, 1356, 4732, 3346, 4560, 6320, 13872, 7506, 21660, 12140, 18900, 21802, 46552, 22008, 53000, 43290, 61668, 49980, 117740, 48450, 153760, 100192, 123552, 129506, 169260, 111420, 312132, 203642, 245544, 195640

Offset: 1

Author

Sebastian Karlsson, Jan 18 2021

Keywords

• nonn

Links

• Sebastian Karlsson, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to lcm's

Crossrefs

Cf. A000010, A006580, A023900, A051193, A057661, A138421.

Programs

Mathematica

a[n_] := Sum[LCM[LCM[n, k], LCM[n, n - k]], {k, 1, n - 1}];
Table[a[n], {n, 1, 40}] (* Robert P. P. McKone, Jan 18 2021 *)

PARI

a(n) = sum(k=1, n-1, lcm(lcm(n, k), lcm(n, n-k))); \\ Michel Marcus, Jan 18 2021

Python

from math import gcd
for n in range(1, 41):
    print(n*sum([k*(n-k)//(gcd(n,k)**2) for k in range(1, n)]), end=', ')

Formula

a(n) = n*Sum_{k=1..n-1} k*(n-k)/gcd(n,k)^2.

a(n) = (1/6)*n*Sum_{d|n} d*(d*phi(d) - A023900(d)).

a(p^e) = (1/6)*p^(e+1)*(p^e-1)*(p^(e+1) + p^(2*e+1) + p^2 + 2*p + 1)/(p^2 + p + 1).

a(prime(n)) = A138421(n). - Michel Marcus, Jan 20 2021