%I #28 Aug 03 2021 02:04:06

%S 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,

%T 359,3,2,79,233,2,7,13,11,5,29,71,139,127,139,2003,5,743,673,593,383,

%U 653,661,251,6389,2833,223,163,37,709,131,41,2203,941,2707,13,1283,383

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

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

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

%C From _Robert G. Wilson v_, Dec 28 2016: (Start)

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

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

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

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

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

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

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

%H Robert G. Wilson v, <a href="/A123487/b123487.txt">Table of n, a(n) for n = 1..205</a>

%H Gareth A. Jones and Alexander K. Zvonkin, <a href="https://arxiv.org/abs/2106.00346">Groups of prime degree and the Bateman-Horn Conjecture</a>, arXiv:2106.00346 [math.GR], 2021.

%t 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 *)

%o (PARI) a(n) = {my(x = 2); while (!isprime(polcyclo(prime(n), x)), x= nextprime(x+1)); x;} \\ _Michel Marcus_, Dec 10 2016

%Y Cf. A123488, A066180, A084732.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Sep 30 2006, Oct 02 2006