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Largest square dividing n!.
12

%I #35 Jul 27 2024 11:14:21

%S 1,1,1,4,4,144,144,576,5184,518400,518400,2073600,2073600,101606400,

%T 914457600,14631321600,14631321600,526727577600,526727577600,

%U 52672757760000,52672757760000,6373403688960000,6373403688960000,917770131210240000,22944253280256000000

%N Largest square dividing n!.

%H Charles R Greathouse IV, <a href="/A055071/b055071.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = A008833(n!).

%F log a(n) ~ n log n. - _Charles R Greathouse IV_, Apr 04 2012

%F a(n) = A055772(n)^2. - _Amiram Eldar_, Jul 26 2024

%p seq(expand(numtheory[nthpow](n!, 2)), n=1..26); # _Peter Luschny_, Apr 03 2013

%t a[n_] := Select[Reverse @ Divisors[n!], IntegerQ[Sqrt[#]] &, 1] // First; a /@ Range[23] (* _Jean-François Alcover_, May 19 2011 *)

%t f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 30] (* _Amiram Eldar_, Jul 26 2024 *)

%o (PARI) a(n)=core(n!,2)[2]^2 \\ _Charles R Greathouse IV_, Apr 04 2012

%o (Python)

%o from math import prod

%o from itertools import count, islice

%o from collections import Counter

%o from sympy import factorint

%o def A055071_gen(): # generator of terms

%o c = Counter()

%o for i in count(1):

%o c += Counter(factorint(i))

%o yield prod(p**(e-(e&1)) for p, e in c.items())

%o A055071_list = list(islice(A055071_gen(),30)) # _Chai Wah Wu_, Jul 27 2024

%Y Cf. A000188, A008833, A055772.

%K nonn

%O 1,4

%A _Labos Elemer_, Jun 13 2000

%E More terms from _James A. Sellers_, Jun 20 2000