A123487 Smallest prime q such that (q^p-1)/(q-1) is prime, where p = prime(n); or 0 if no such prime q exists.

2, 2, 2, 2, 5, 2, 2, 2, 113, 151, 2, 61, 53, 89, 5, 307, 19, 2, 491, 3, 11, 271, 41, 2, 271, 359, 3, 2, 79, 233, 2, 7, 13, 11, 5, 29, 71, 139, 127, 139, 2003, 5, 743, 673, 593, 383, 653, 661, 251, 6389, 2833, 223, 163, 37, 709, 131, 41, 2203, 941, 2707, 13, 1283, 383

Offset: 1

Author

Alexander Adamchuk, Sep 30 2006, Oct 02 2006

Keywords

• nonn

Comments

Corresponding primes (q^p-1)/(q-1) are listed in A123488.

a(n) coincides with A066180(n) when A066180(n) is prime or 0.

From Robert G. Wilson v, Dec 28 2016: (Start)

Conjecture: Never is a(n) equal to 0.

Records: 2, 5, 113, 151, 307, 491, 2003, 6389, 7883, 11813, 18587, 31721, 40763, ... ;

First occurrence of the k_th prime: 1, 20, 5, 32, 21, 33, 81, 17, ... ;

Positions where two occurs: 1, 2, 3, 4, 6, 7, 8, 11, 18, 24, 28, 31, 98, 111, ... ;

Positions where three occurs: 20, 27, 100, 182, ... ;

Positions where five occurs: 5, 15, 35, 42, 114, 158, ... ; etc. (End)

Jones & Zvonkin conjecture (as did Robert G. Wilson v above) that a(n) > 0 for all n. - Charles R Greathouse IV, Jul 23 2021

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..205
• Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn Conjecture, arXiv:2106.00346 [math.GR], 2021.

Crossrefs

Cf. A123488, A066180, A084732.

Programs

Mathematica

f[n_] := NestWhile[NextPrime, 2, ! PrimeQ[Cyclotomic[Prime[n], #]] &]; Array[f, 63](* Davin Park, Dec 28 2016 and Robert G. Wilson v, Dec 28 2016 *)

PARI

a(n) = {my(x = 2); while (!isprime(polcyclo(prime(n), x)), x= nextprime(x+1)); x;} \\ Michel Marcus, Dec 10 2016