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A055071 Largest square dividing n!.

%I A055071 #36 Jul 02 2025 16:01:59
%S A055071 1,1,1,4,4,144,144,576,5184,518400,518400,2073600,2073600,101606400,
%T A055071 914457600,14631321600,14631321600,526727577600,526727577600,
%U A055071 52672757760000,52672757760000,6373403688960000,6373403688960000,917770131210240000,22944253280256000000
%N A055071 Largest square dividing n!.
%H A055071 Charles R Greathouse IV, <a href="/A055071/b055071.txt">Table of n, a(n) for n = 1..500</a>
%F A055071 a(n) = A008833(n!).
%F A055071 log a(n) ~ n log n. - _Charles R Greathouse IV_, Apr 04 2012
%F A055071 a(n) = A055772(n)^2. - _Amiram Eldar_, Jul 26 2024
%p A055071 seq(expand(numtheory[nthpow](n!, 2)), n=1..26); # _Peter Luschny_, Apr 03 2013
%t A055071 a[n_] := Select[Reverse @ Divisors[n!], IntegerQ[Sqrt[#]] &, 1] // First; a /@ Range[23] (* _Jean-François Alcover_, May 19 2011 *)
%t A055071 f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 30] (* _Amiram Eldar_, Jul 26 2024 *)
%o A055071 (PARI) a(n)=core(n!,2)[2]^2 \\ _Charles R Greathouse IV_, Apr 04 2012
%o A055071 (Python)
%o A055071 from math import prod
%o A055071 from itertools import count, islice
%o A055071 from collections import Counter
%o A055071 from sympy import factorint
%o A055071 def A055071_gen(): # generator of terms
%o A055071     c = Counter()
%o A055071     for i in count(1):
%o A055071         c += Counter(factorint(i))
%o A055071         yield prod(p**(e-(e&1)) for p, e in c.items())
%o A055071 A055071_list = list(islice(A055071_gen(),30)) # _Chai Wah Wu_, Jul 27 2024
%Y A055071 Cf. A000188, A008833, A055772.
%K A055071 nonn
%O A055071 1,4
%A A055071 _Labos Elemer_, Jun 13 2000
%E A055071 More terms from _James Sellers_, Jun 20 2000