A055071 Largest square dividing n!.

1, 1, 1, 4, 4, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400, 914457600, 14631321600, 14631321600, 526727577600, 526727577600, 52672757760000, 52672757760000, 6373403688960000, 6373403688960000, 917770131210240000, 22944253280256000000

Offset: 1

Author

Labos Elemer, Jun 13 2000

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000188, A008833, A055772.

Programs

Maple

seq(expand(numtheory[nthpow](n!, 2)), n=1..26); # Peter Luschny, Apr 03 2013

Mathematica

a[n_] := Select[Reverse @ Divisors[n!], IntegerQ[Sqrt[#]] &, 1] // First; a /@ Range[23] (* Jean-François Alcover, May 19 2011 *)
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 30] (* Amiram Eldar, Jul 26 2024 *)

PARI

a(n)=core(n!,2)[2]^2 \\ Charles R Greathouse IV, Apr 04 2012

Python

from math import prod
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A055071_gen(): # generator of terms
    c = Counter()
    for i in count(1):
        c += Counter(factorint(i))
        yield prod(p**(e-(e&1)) for p, e in c.items())
A055071_list = list(islice(A055071_gen(),30)) # Chai Wah Wu, Jul 27 2024

Formula

a(n) = A008833(n!).

log a(n) ~ n log n. - Charles R Greathouse IV, Apr 04 2012

a(n) = A055772(n)^2. - Amiram Eldar, Jul 26 2024

Extensions

More terms from James Sellers, Jun 20 2000