

A093804


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


4




OFFSET

1,1


COMMENTS

Or, numbers n such that Sum_{dn} d! is prime.
The prime 26951 from A002981 (n!+1 is prime) is a member since Sum_{dn} 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


LINKS

Table of n, a(n) for n=1..9.
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 BC2012) 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.


EXAMPLE

Sum_{d3} d! = 1! + 3! = 7 is prime, so 3 is a member.


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 *)


PROG

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


CROSSREFS

Cf. A062363, A002981, A038507, A088332, A103317, A103319.
Sequence in context: A006497 A038912 A019361 * A084121 A088747 A231574
Adjacent sequences: A093801 A093802 A093803 * A093805 A093806 A093807


KEYWORD

nonn,more,hard


AUTHOR

Jason Earls, May 19 2004


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


STATUS

approved



