A082995 Distance from n!+1 to next larger square.

2, 1, 2, 0, 0, 8, 0, 80, 728, 224, 323, 39168, 82943, 176399, 215295, 3444735, 26167683, 114349224, 255004928, 1158920360, 11638526760, 42128246888, 191052974115, 97216010328, 2430400258224, 1553580508515, 4666092737475

Offset: 1

Author

Jason Earls, May 29 2003

Keywords

• nonn

Comments

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?

Examples

			a(5)=0 because 5!+1 is a square.
a(8)=80 because 8!+1 = 40321 and the next larger square is 40401, so 40401-40321 = 80.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..808
• P. Leyland, Solutions to n!+1=m^2.

Crossrefs

Cf. A038507, A087374.

Programs

Mathematica

a[n_] := Ceiling @ Sqrt[(f = n! + 1)]^2 - f; Array[a, 27] (* Amiram Eldar, Dec 14 2019 *)

PARI

for(k=1,27,print1(ceil(sqrt(k!+1))^2-(k!+1),", ")) \\ Hugo Pfoertner, Dec 14 2019