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Squarefree part of n!: n! divided by its largest square divisor.
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%I #38 Nov 17 2024 21:38:35

%S 1,2,6,6,30,5,35,70,70,7,77,231,3003,858,1430,1430,24310,12155,230945,

%T 46189,969969,176358,4056234,676039,676039,104006,312018,44574,

%U 1292646,1077205,33393355,66786710,2203961430,64822395,90751353,90751353

%N Squarefree part of n!: n! divided by its largest square divisor.

%C Smallest number such that n!*a(n) is a square.

%H Charles R Greathouse IV, <a href="/A055204/b055204.txt">Table of n, a(n) for n = 1..3386</a> (next term has 1001 digits)

%H Kevin A. Broughan, <a href="https://www.emis.de/journals/INTEGERS/papers/c10/c10.Abstract.html">Asymptotic Order of the Square-free Part of N!</a>, Integers, 2 (2002), Article A.10.

%H Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7542">On the h-th free part of the factorial</a>, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

%F a(n) = A007913(n!) = n!/A055071(n) = A000142(n)/A055071(n).

%F log a(n) ~ n log 2. - _Charles R Greathouse IV_, Apr 03 2012

%F sqrt(n!) = A055772(n) * sqrt(a(n)). - _Alonso del Arte_, Feb 16 2015

%e 10! = 518400*7 = 7*(720)^2, so a(10) = 7.

%t f[p_, e_] := p^Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* _Amiram Eldar_, Sep 01 2024 *)

%t a[n_] := Block[{fi = Transpose@ FactorInteger[n!]}, Times @@ (fi[[1]]^Mod[fi[[2]], 2])]; Array[a, 40] (* _Robert G. Wilson v_, Nov 17 2024 *)

%o (PARI) a(n)=core(n!) \\ _Charles R Greathouse IV_, Apr 03 2012

%Y Cf. A000142, A007913, A055071, A249831.

%K nonn,changed

%O 1,2

%A _Labos Elemer_, Jun 19 2000