A100596 Numbers k such that (prime(k)-1)! + prime(k)^10 is prime.

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

Offset: 1

Author

Jonathan Vos Post, Nov 30 2004

Keywords

• nonn
• hard
• more

Comments

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.

a(7) > 600. - Jinyuan Wang, Apr 10 2020

a(8) > 2700. - Michael S. Branicky, Jul 03 2024

Examples

			a(1) = 2 because (prime(2)-1)! + prime(2)^10 = (3-1)! + 3^10 = 59051 is the smallest prime of that form.
a(2) = 8 because (prime(8)-1)! + prime(8)^10 = (19-1)! + 19^10 = 6408504771985801 is the 2nd smallest prime of that form.
		

Crossrefs

Cf. A100595, A100858.

Programs

Mathematica

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

Python

from math import factorial
from sympy import isprime, prime
def afind(limit, startat=1):
    for k in range(startat, limit+1):
        s = str(k)
        pk = prime(k)
        if isprime( factorial(pk-1) + pk**10 ):
            print(k, end=", ")
afind(100) # Michael S. Branicky, Nov 30 2021

Formula

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

Extensions

a(6) from Jinyuan Wang, Apr 10 2020

a(7) from Michael S. Branicky, Nov 30 2021