A384452 a(n) is the sum of squares of the unitary divisors of n!.

1, 5, 50, 650, 16900, 547924, 27396200, 1746641000, 139773881000, 13460683752200, 1642203417768400, 236441876606410000, 40195119023089700000, 7723888546922636420000, 1735183690969722609168800, 444206919394766468845892000, 128820006624482275965308680000, 41737604550102658693597600532800

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Project Euler, Problem 429: Sum of Squares of Unitary Divisors.

Wikipedia, Legendre's formula.

Formula

a(n) = Sum_{d|n!} (d^2 if gcd(d,n!//d) = 1).

a(n) = Product_{p <= n, p prime} (p^(2*f(n,p)))+1 with f(n,p) = f(floor(n/p)) + floor(n/p) and f(0,p) = 0 where f(n,p) is equivalent to the Legendre formula.

a(n) = A034676(n!).

Mathematica

f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 18] (* Amiram Eldar, Jun 02 2025 *)

Prog

(Python)
from sympy import nextprime
def f(n, p):
  if n==0: return 0
  return f(n//p, p) + n//p
def a(n):
  s, p = 1, 2
  while p<=n:
    s *= p**(f(n, p)<<1)+1
    p = nextprime(p)
  return s
print([a(n) for n in range(1, 19)])
(PARI) row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); } \\ A077610
a(n) = norml2(row(n!)); \\ Michel Marcus, Jun 02 2025

Crossrefs

Cf. A000142, A034676, A064028, A077610.

Sequence in context: A047736 A185272 A141316 * A217794 A347022 A381262

Adjacent sequences: A384449 A384450 A384451 * A384453 A384454 A384455

Keyword

nonn

Author

Darío Clavijo, Jun 02 2025

Status

approved