%I #17 Dec 14 2019 08:10:02
%S 2,1,2,0,0,8,0,80,728,224,323,39168,82943,176399,215295,3444735,
%T 26167683,114349224,255004928,1158920360,11638526760,42128246888,
%U 191052974115,97216010328,2430400258224,1553580508515,4666092737475
%N Distance from n!+1 to next larger square.
%C The only known values of n such that n!+1 is a perfect square are 4, 5 and 7. Paul Leyland, et al. have found no other solutions for n <= 1 million (see link). For 1 <= n <= 11, n!+1 is within 1000 of being a square. Is there another n such that n!+1 <= "1000 away" from being a perfect square?
%H Amiram Eldar, <a href="/A082995/b082995.txt">Table of n, a(n) for n = 1..808</a>
%H P. Leyland, <a href="http://www.leyland.vispa.com/numth/misc_cnt/oddities.htm">Solutions to n!+1=m^2</a>.
%e a(5)=0 because 5!+1 is a square.
%e a(8)=80 because 8!+1 = 40321 and the next larger square is 40401, so 40401-40321 = 80.
%t a[n_] := Ceiling @ Sqrt[(f = n! + 1)]^2 - f; Array[a, 27] (* _Amiram Eldar_, Dec 14 2019 *)
%o (PARI) for(k=1,27,print1(ceil(sqrt(k!+1))^2-(k!+1),", ")) \\ _Hugo Pfoertner_, Dec 14 2019
%Y Cf. A038507, A087374.
%K nonn
%O 1,1
%A _Jason Earls_, May 29 2003