OFFSET
1,1
COMMENTS
1) Necessarily a(n) is not divisible by 2, 3, 5, 7.
2) Sequence is conjectured to be infinite.
3) It is conjectured that an infinite number of terms are primes.
4) Note that sequence contains a(k), a(k+1) prime twin pairs, first are (59,61), (461,463), (827,829), (1319,1321).
5) It is conjectured that an infinite number of a(n) are squares, first are 121=11^2, 169=13^2, 529=23^2, 841=29^2, 961=31^2, 1681=41^2, ...
6) m!+k^2=n^2 are the generalized Brown number triples (m,k,n).
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, p. 193, 1994
I. Niven, H. S. Zuckerman and H. L. Montgomery: An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books, 1991
David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
FORMULA
7! + a(n)^2 = prime.
EXAMPLE
1) 7!+1=71^2, (7, 71) is the largest (of three) Brown pairs; Erdos conjectured that there are no others.
2) 7!+3^2=5049= 3^3 * 11 * 17, 7!+5^2=5065 = 5 * 1013, 7!+7^2=5089 = 7 * 727, 7!+9^2=5121 = 3^2 * 569, 7!+11^2=5161 = 13 * 397.
3) 7!+13^2=5209 prime, so a(1)=13.
MATHEMATICA
With[{s = 7!}, Select[Range[600], PrimeQ[#^2 + s] &]] (* Harvey P. Dale, Jun 17 2015 *)
PROG
(PARI) isok(n) = isprime(n^2+7!); \\ Michel Marcus, Jul 23 2013; corrected Jun 14 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 05 2009
EXTENSIONS
Edited by N. J. A. Sloane, Apr 05 2009
STATUS
approved