A093804 Primes p such that p! + 1 is also prime.

2, 3, 11, 37, 41, 73, 26951, 110059, 150209

Offset: 1

Author

Jason Earls, May 19 2004

Keywords

• nonn
• more
• hard

Comments

Or, numbers n such that Sum_{d|n} d! is prime.

The prime 26951 from A002981 (n!+1 is prime) is a member since Sum_{d|n} d! = n!+1 if n is prime. - Jonathan Sondow, Jan 30 2005

a(n) are the primes in A002981[n] = {0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ...} Numbers n such that n! + 1 is prime. Corresponding primes of the form p! + 1 are listed in A103319[n] = {3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, ...}. - Alexander Adamchuk, Sep 23 2006

Examples

			Sum_{d|3} d! = 1! + 3! = 7 is prime, so 3 is a member.
		

Links

• Chris K. Caldwell, The List of Largest Known Primes, 110059! + 1
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
• Apoloniusz Tyszka, A common approach to the problem of the infinitude of twin primes, primes of the form n! + 1, and primes of the form n! - 1, 2018.
• Apoloniusz Tyszka, A new approach to solving number theoretic problems, 2018.

Crossrefs

Cf. A062363, A002981, A038507, A088332, A103317, A103319.

Programs

Maple

seq(`if`(isprime(ithprime(n)!+1), ithprime(n), NULL),n=1..25); # Nathaniel Johnston, Jun 28 2011

Mathematica

Select[Prime[Range[5! ]],PrimeQ[ #!+1]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

PARI

isok(n) = ispseudoprime(n) && ispseudoprime(n!+1); \\ Jinyuan Wang, Jan 20 2020

Extensions

One more term from Alexander Adamchuk, Sep 23 2006

a(8)=110059 (found on Jun 11 2011, by PrimeGrid), added by Arkadiusz Wesolowski, Jun 28 2011

a(9)=150209 (found on Jun 09 2012, by Rene Dohmen), added by Jinyuan Wang, Jan 20 2020