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

1,1

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?

Links

Amiram Eldar, Table of n, a(n) for n = 1..808

P. Leyland, Solutions to n!+1=m^2.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A038507, A087374.

Sequence in context: A317741 A317902 A317871 * A079549 A143374 A277899

Adjacent sequences: A082992 A082993 A082994 * A082996 A082997 A082998

Keyword

nonn

Author

Jason Earls, May 29 2003

Status

approved