A068869 Smallest number k such that n! + k is a square.

0, 2, 3, 1, 1, 9, 1, 81, 729, 225, 324, 39169, 82944, 176400, 215296, 3444736, 26167684, 114349225, 255004929, 1158920361, 11638526761, 42128246889, 191052974116, 97216010329, 2430400258225, 1553580508516, 4666092737476, 565986718738441, 2137864362693921

Offset: 1

Author

Amarnath Murthy, Mar 13 2002

Keywords

• nonn

Comments

Observation: for n < 2000, only for n = 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16 is a(n) a square (see A360210).

According to my conjecture that n! + n^2 != m^2 for n >= 1, m >= 0 (see A004664), for all terms: a(n) != n^2. - Alexander R. Povolotsky, Oct 06 2008

There are two cases: a(n) > sqrt(n!) in A182203 and a(n) < sqrt(n!) in A182204. - Artur Jasinski, Apr 13 2012

Examples

			a(6) = 9 as 6! + 9 = 729 is a square.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..100

Crossrefs

Cf. A038202, A048761, A055228, A066857, A360210.

Cf. A182203, A182204.

Programs

Mathematica

Table[ Ceiling[ Sqrt[n! ]]^2 - n!, {n, 1, 28}]

PARI

A068869(n)=(sqrtint(n!-1)+1)^2-n!  \\ M. F. Hasler, Apr 01 2012

Python

from math import factorial, isqrt
def a(n): return (isqrt((f:=factorial(n))-1)+1)**2 - f
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Jan 30 2023

Formula

a(n) = A055228(n)^2 - n! = ceiling(sqrt(n!))^2 - n! = A048761(n!) - n!.

a(n) <= A038202(n)^2, with equality for the n listed in the first comment. - M. F. Hasler, Apr 01 2012

Extensions

More terms from Vladeta Jovovic, Mar 21 2002

Edited by Robert G. Wilson v and N. J. A. Sloane, Mar 22 2002