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%I #47 Apr 03 2023 10:36:10
%S 2,3,11,37,41,73,26951,110059,150209
%N Primes p such that p! + 1 is also prime.
%C Or, numbers n such that Sum_{d|n} d! is prime.
%C 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
%C 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
%H Chris K. Caldwell, The List of Largest Known Primes, <a href="https://t5k.org/primes/page.php?id=100445">110059! + 1</a>
%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
%H Apoloniusz Tyszka, <a href="https://hal.archives-ouvertes.fr/hal-01614087v5/document">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</a>, 2018.
%H Apoloniusz Tyszka, <a href="https://philarchive.org/archive/TYSDASv8">A new approach to solving number theoretic problems</a>, 2018.
%e Sum_{d|3} d! = 1! + 3! = 7 is prime, so 3 is a member.
%p seq(`if`(isprime(ithprime(n)!+1), ithprime(n), NULL),n=1..25); # _Nathaniel Johnston_, Jun 28 2011
%t Select[Prime[Range[5! ]],PrimeQ[ #!+1]&] (* _Vladimir Joseph Stephan Orlovsky_, Nov 17 2009 *)
%o (PARI) isok(n) = ispseudoprime(n) && ispseudoprime(n!+1); \\ _Jinyuan Wang_, Jan 20 2020
%Y Cf. A062363, A002981, A038507, A088332, A103317, A103319.
%K nonn,more,hard
%O 1,1
%A _Jason Earls_, May 19 2004
%E One more term from _Alexander Adamchuk_, Sep 23 2006
%E a(8)=110059 (found on Jun 11 2011, by PrimeGrid), added by _Arkadiusz Wesolowski_, Jun 28 2011
%E a(9)=150209 (found on Jun 09 2012, by Rene Dohmen), added by _Jinyuan Wang_, Jan 20 2020
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