A159082 Numbers whose squares added to 7! are prime.

13, 23, 29, 59, 61, 73, 97, 101, 103, 109, 121, 127, 149, 169, 187, 191, 199, 221, 227, 251, 257, 263, 277, 299, 307, 317, 319, 331, 341, 367, 373, 383, 389, 397, 403, 407, 409, 433, 449, 451, 461, 463, 467, 491, 493, 499, 517, 527, 529, 533, 551, 563, 571

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

Cf. A038202, A158979.

Sequence in context: A255581 A061967 A249350 * A303576 A228324 A171122

Adjacent sequences: A159079 A159080 A159081 * A159083 A159084 A159085

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