A307590 a(n) is the smallest base b such that q = b^n - b^m + 1 is prime, where m = A276976(n).

2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 14, 11, 2, 11, 29, 11, 5, 19, 14, 6, 27, 2, 3, 21, 8, 7, 10, 3, 4, 2, 14, 3, 5, 106, 3, 2, 44, 4, 3, 43, 4, 4, 21, 6, 16, 25, 41, 3, 12, 14, 10, 2, 3, 81, 28, 27, 66, 37, 17, 61, 5, 22, 12, 179, 197, 49, 2, 132, 178, 11

Offset

1,1

Comments

If p is a prime, then a(p) is the smallest base b such that q = b^p - b + 1 is prime. These primes q == 1 (mod p) by Fermat's Little Theorem. Note that if p is a prime, then a(p) = 2 if and only if 2^p - 1 is prime, so p is a Mersenne exponent in A000043. Composite numbers n such that a(n) = 2 are 4, 6, 8, 10, 12, 14, 16, 22, 39, 45, 76, ... Cf. composite terms in A307625. Except 8, are these the same numbers?

a(80) does not exist because A276976(80) = 4 and b^8-b^4+1 is a factor of b^80-b^4+1. Similarly, a(n) also does not exist for n = 84, 160, 312, 320, 400, 588, 640, 684, 800, ... - Giovanni Resta, Apr 24 2019

Links

Table of n, a(n) for n=1..79.

Formula

q == 1 (mod n).

Example

a(9) = 5 so the number 5^9 - 5^3 + 1 is a prime q == 1 (mod 9).

Mathematica

fQ[n_, m_] := AllTrue[Range[2, n - 1], PowerMod[#, m, n] == PowerMod[#, n, n] &]; f[1] = 0; f[2] = 1; f[n_] := Module[{m = 0}, While[!fQ[n, m], m++]; m]; a[n_] := Module[{b = 2, m = f[n]}, While[!PrimeQ[b^n - b^m + 1], b++]; b]; Array[a, 79] (* Amiram Eldar, Apr 19 2019 *)

Prog

(PARI) a276976(n)=if(n<3, return(n-1)); forstep(m=1, n, n%2+1, for(b=0, n-1, if(Mod(b, n)^m-Mod(b, n)^n, next(2))); return(m)); \\ A276976
a(n) = my(b=2); while (!isprime(b^n - b^a276976(n) + 1), b++); b; \\ Michel Marcus, Apr 21 2019

Crossrefs

Cf. A000043, A276976, A307625.

Sequence in context: A104274 A008857 A244463 * A307987 A047935 A365882

Adjacent sequences: A307587 A307588 A307589 * A307591 A307592 A307593

Keyword

nonn

Author

Thomas Ordowski, Apr 19 2019

Extensions

More terms from Amiram Eldar, Apr 19 2019

Status

approved