A083397 Largest prime factor of n! + k where k is the least positive integer such that n! + k is a square.

0, 2, 3, 5, 11, 3, 71, 67, 67, 127, 13, 509, 137, 37, 71471, 71471, 409993, 941351, 24419, 287093, 7147792819, 110647261, 80392811773, 4716679469, 4716679469, 323905128133, 8392290961, 551615338229, 34178276390953, 73669621631

Offset: 1

Author

Jason Earls, Jun 06 2003

Keywords

• nonn

Comments

For n > 1, n! cannot be a perfect square. Proof: All exponents of the prime factors of a square are even. But in the factorization of n! at least one of the primes will appear only once due to Bertrand's Postulate which says there is always a prime between m and 2m.

Examples

			a(9)=67 because 9!+729 = 363609 = 3^4*67^2 is a square with largest prime factor of 67.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A068869.

Programs

Mathematica

Join[{0},Table[FactorInteger[(Floor[Sqrt[n!]]+1)^2][[-1,1]],{n,2,30}]] (* Harvey P. Dale, Jan 04 2012 *)