A038202 Least k such that n! + k^2 is a square.

1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856, 10080, 46848, 210240, 400320, 652848, 3991680, 27528402, 32659200, 163296000, 1143463200, 1305467240, 6840489600, 9453465438, 337082683248, 163425485250, 8376514506360, 8440230839040, 5088099594240

Offset: 4

Author

David W. Wilson

Keywords

• nonn

Comments

Let f = n!/4 and let x be the largest divisor of f such that x < sqrt(f). Then a(n) = f/x - x. The greatest k such that n! + k^2 is a square is f-1. The number of k for which n! + k^2 is a square is A038548(n). - T. D. Noe, Nov 02 2004

For greatest k such that n! + k^2 is a square see A181892; for numbers x such that n! + k^2 = x^2 see A181896. - Artur Jasinski, Mar 31 2012

Links

• Sudipta Mallick, Table of n, a(n) for n = 4..58
• Eric Weisstein's World of Mathematics, Brocard's Problem

Crossrefs

Cf. A038548 (number of divisors of n that are at most sqrt(n)), A068869.

Cf. A181892, A181896.

Programs

Mathematica

Table[f=n!/4; x=Max[Select[Divisors[f], #<=Sqrt[f]&]]; f/x-x, {n, 4, 20}] (* T. D. Noe, Nov 02 2004 *)

PARI

a(n) = my(k=0); while(!issquare(n!+k^2), k++); k; \\ Michel Marcus, Sep 16 2018

Extensions

a(30)-a(34) from Jon E. Schoenfield, Sep 15 2018