The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A307590 a(n) is the smallest base b such that q = b^n - b^m + 1 is prime, where m = A276976(n). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 7 19:59 EDT 2024. Contains 375749 sequences. (Running on oeis4.)