A100595 - OEIS

%I #17 Jul 03 2024 16:42:51

%S 9,10,17,137

%N Numbers k such that (prime(k)-1)! + prime(k)^9 is prime.

%C There are no more such k up to k=150. Computed in collaboration with _Ray Chandler_.

%C a(5) > 1000. - _Jinyuan Wang_, Apr 11 2020

%C a(5) > 2700. - _Michael S. Branicky_, Jul 03 2024

%F Numbers k such that (prime(k)-1)! + prime(k)^9 is prime, where prime(k) is the k-th prime.

%e a(1) = 9 because (prime(9)-1)! + prime(9)^9 = (23-1)! + 23^9 = 1124000729578760341463 is the smallest prime of this form.

%e a(2) = 10 because (prime(10)-1)! + prime(10)^9 = (29-1)! + 29^9 = 304888344611713875008649975869 is the 2nd smallest prime of this form.

%e a(3) = 17, but prime(17) = 59 yields a number that would take 2 full lines of this page; and a(4) = 137 because prime(137) = 773 yields a prime of this form which is 1975 digits long. Note also that 773 = prime(137) = prime(prime(34)).

%t lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^9], AppendTo[lst, n]], {n, 12^2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 08 2008 *)

%o (PARI) is(k) = ispseudoprime((prime(k)-1)! + prime(k)^9); \\ _Jinyuan Wang_, Apr 11 2020

%Y Cf. A100596, A100598, A100858.

%K nonn,hard,more

%O 1,1

%A _Jonathan Vos Post_, Nov 30 2004