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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn Conjecture, arXiv:2106.00346 [math.GR], 2021.
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 *)
PROG
(PARI) a(n) = {my(x = 2); while (!isprime(polcyclo(prime(n), x)), x= nextprime(x+1)); x; } \\ Michel Marcus, Dec 10 2016
CROSSREFS
Sequence in context: A341444 A084731 A066180 * A130325 A362034 A154097
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Sep 30 2006, Oct 02 2006
STATUS
approved

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Last modified July 22 19:00 EDT 2024. Contains 374540 sequences. (Running on oeis4.)