cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Thomas Ordowski, Apr 19 2019

Keywords

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

Examples

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

Crossrefs

Programs

  • 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 *)
  • 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

Formula

q == 1 (mod n).

Extensions

More terms from Amiram Eldar, Apr 19 2019
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