A125973 Smallest k such that k^n + k^(n-1) - 1 is prime.

2, 2, 2, 2, 2, 3, 2, 2, 14, 4, 7, 2, 38, 6, 7, 3, 4, 10, 2, 9, 74, 6, 10, 7, 4, 61, 20, 4, 5, 9, 6, 16, 6, 8, 2, 9, 4, 10, 2, 48, 44, 163, 9, 2, 95, 3, 27, 70, 6, 26, 57, 9, 6, 8, 207, 2, 27, 15, 45, 7, 69, 199, 55, 16, 2, 5, 12, 43, 137, 39, 9, 57, 5, 20, 4, 115, 2, 103, 45, 15, 20, 109

Offset: 1

Author

Artur Jasinski, Dec 14 2006

Keywords

• nonn

Comments

The polynomial x^n + x^(n-1) - 1 is irreducible over the rationals (see Ljunggren link), so the Bunyakovsky conjecture implies that a(n) always exists. - Robert Israel, Nov 16 2016

Examples

			Consider n = 6. k^6 + k^5 - 1 evaluates to 1, 95, 971 for k = 1, 2, 3. Only the last of these numbers is prime, hence a(6) = 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..800
• W. Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand. 8 (1960) 65-70.
• Wikipedia, Bunyakovsky conjecture.

Crossrefs

Cf. A000040, A045546, A125881-A125885, A125965-A125972, A126017.

Cf. A091997 (n such that a(n)=2).

Programs

Maple

f:= proc(n) local k;
for k from 2 do if isprime(k^n+k^(n-1)-1) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 16 2016

Mathematica

a[n_] := For[k = 2, True, k++, If[PrimeQ[k^n + k^(n-1) - 1], Return[k]]];
Array[a, 100] (* Jean-François Alcover, Feb 26 2019 *)

PARI

{m=82;for(n=1,m,k=1;while(!isprime(k^n+k^(n-1)-1),k++);print1(k,","))} \\ Klaus Brockhaus, Dec 17 2006

Extensions

Edited and extended by Klaus Brockhaus, Dec 17 2006