A100596 - OEIS

%I #21 Dec 27 2024 13:49:33

%S 2,8,15,33,52,205,751

%N Numbers k such that (prime(k)-1)! + prime(k)^10 is prime.

%C k = {2, 8, 15, 33, 52, 205} yields primes p(k) = {3, 19, 47, 137, 239, 1259}. There are no more such k up to k=150. Computed in collaboration with _Ray Chandler_.

%C a(7) > 600. - _Jinyuan Wang_, Apr 10 2020

%C a(8) > 2700. - _Michael S. Branicky_, Jul 03 2024

%F Primes of the form (prime(k)-1)! + prime(k)^10, where prime(k) is the k-th prime.

%e a(1) = 2 because (prime(2)-1)! + prime(2)^10 = (3-1)! + 3^10 = 59051 is the smallest prime of that form.

%e a(2) = 8 because (prime(8)-1)! + prime(8)^10 = (19-1)! + 19^10 = 6408504771985801 is the 2nd smallest prime of that form.

%t lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^10], AppendTo[lst, n]], {n, 10^2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 08 2008 *)

%t Select[Range[250],PrimeQ[(Prime[#]-1)!+Prime[#]^10]&] (* The program generates the first 6 terms of the sequence. *) (* _Harvey P. Dale_, Dec 27 2024 *)

%o (Python)

%o from math import factorial

%o from sympy import isprime, prime

%o def afind(limit, startat=1):

%o     for k in range(startat, limit+1):

%o         s = str(k)

%o         pk = prime(k)

%o         if isprime( factorial(pk-1) + pk**10 ):

%o             print(k, end=", ")

%o afind(100) # _Michael S. Branicky_, Nov 30 2021

%Y Cf. A100595, A100858.

%K nonn,hard,more

%O 1,1

%A _Jonathan Vos Post_, Nov 30 2004

%E a(6) from _Jinyuan Wang_, Apr 10 2020

%E a(7) from _Michael S. Branicky_, Nov 30 2021