A093324 a(n) is the smallest natural number m such that n^m + m is prime.

2, 1, 1, 2, 1, 7954, 1, 34, 101, 2, 1, 1181716, 1, 54, 17, 2, 1, 1080, 1, 57910, 9, 2, 1, 202, 2075, 5538, 3

Offset: 0

Author

Farideh Firoozbakht, May 11 2004

Keywords

• nonn
• hard
• more

Comments

a(n)=1 iff n+1 is prime. For n>1 n+a(n) is odd (for n>1 a(n) is odd iff n is even). a(11) is greater than 7000. a(m) for m = 12, 13, 14, 15, 16, 17 and 18 are 1, 54, 17, 2, 1, 1080, 1. 5^7954 + 7954 is a 5560-digit probable prime.

a(11) is greater than 8651. - Robert G. Wilson v, May 24 2004

a(11) is greater than 20165, so every prime of the form 11^m + m has more than 21000 digits. - Farideh Firoozbakht, Jun 10 2004

a(11) is greater than 80000. - T. D. Noe, Mar 07 2007

a(11) is greater than 190000. - T. D. Noe, Jul 31 2008

Examples

			a(3)=2 because 3^2 + 2 is prime and 3^1 + 1 is composite.
		

Links

• Henri Lifchitz & Renaud Lifchitz, 11^1181716+1181716, a(11).
• Henri Lifchitz & Renaud Lifchitz, 19^57910+57910, a(19).

Programs

Magma

function A093324(n)
  t:=0;
    while not IsPrime(n^t + t) do
      t+:=1;
    end while;
  return t;
end function;
[A093324(n): n in [0..10]]; // G. C. Greubel, Aug 10 2023

Mathematica

a[n_]:= (For[m=1, !PrimeQ[n^m+m], m++]; m); Do[Print[a[n]], {n,0,10}]

Python

from sympy import isprime
def a(n):
  m = 0
  while not isprime(n**m + m): m += 1
  return m
for n in range(11):
  print(a(n), end=", ") # Michael S. Branicky, Feb 01 2021

Extensions

a(11)-a(26) from Kellen Shenton, Aug 14 2023