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%I #57 Apr 29 2021 01:42:44
%S 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,
%T 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,
%U 14,10,2,3,81,28,27,66,37,17,61,5,22,12,179,197,49,2,132,178,11
%N a(n) is the smallest base b such that q = b^n - b^m + 1 is prime, where m = A276976(n).
%C 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?
%C 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
%F q == 1 (mod n).
%e a(9) = 5 so the number 5^9 - 5^3 + 1 is a prime q == 1 (mod 9).
%t 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 *)
%o (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
%o a(n) = my(b=2); while (!isprime(b^n - b^a276976(n) + 1), b++); b; \\ _Michel Marcus_, Apr 21 2019
%Y Cf. A000043, A276976, A307625.
%K nonn
%O 1,1
%A _Thomas Ordowski_, Apr 19 2019
%E More terms from _Amiram Eldar_, Apr 19 2019