A103794 Smallest number b such that b^prime(n) - (b-1)^prime(n) is prime.

2, 2, 2, 2, 6, 2, 2, 2, 6, 3, 2, 40, 7, 5, 13, 3, 3, 2, 7, 18, 47, 8, 6, 2, 26, 3, 42, 2, 13, 8, 2, 8, 328, 8, 9, 45, 27, 13, 76, 15, 52, 111, 5, 15, 50, 287, 16, 5, 40, 23, 110, 368, 23, 68, 28, 96, 81, 150, 3, 143, 4, 12, 403, 4, 45, 11, 83, 21, 96, 5, 109, 350, 128, 304, 38, 4, 163

Offset: 1

Author

Lei Zhou, Feb 24 2005

Keywords

• nonn

Comments

Conjecture: sequence is defined for all positive indices.

For p=prime(n), Eisenstein's irreducibility criterion can be used to show that the polynomial (x+1)^p-x^p is irreducible, which is a necessary (but not sufficient) condition for a(n) to exist. - T. D. Noe, Dec 05 2005

Examples

			2^prime(1)-1^prime(1)=3 is prime, so a(1)=2;
2^prime(5)-1^prime(5)=2047 has a factor of 23;
...
6^prime(5)-5^prime(5)=313968931 is prime, so a(5)=6;
		

Crossrefs

Cf. A103795, A066180, A058013, A222119.

Programs

Maple

f:= proc(n) local p,b;
  p:= ithprime(n);
  for b from 2 do
    if isprime(b^p - (b-1)^p) then return b fi
  od
end proc:
map(f, [$1..80]); # Robert Israel, Jun 04 2024

Mathematica

Do[p=Prime[k]; n=2; nm1=n-1; cp=n^p-nm1^p; While[ !PrimeQ[cp], n=n+1; nm1=n-1; cp=n^p-nm1^p]; Print[n], {k, 1, 200}]

Formula

a(n) = A222119(n) + 1. - Ray Chandler, Feb 26 2017