A353101 Least b > 1 such that (b^(prime(n)^2) - 1)/(b^prime(n) - 1) is prime.

2, 2, 22, 2, 43, 24, 315, 38, 54, 265, 605, 61, 697, 306, 1153, 370, 2, 10688, 3075, 2338, 1153, 3243, 130, 2301, 315, 200, 1155, 14739, 4591, 2230, 263, 6665, 250, 10520, 2228, 3699, 1126, 8925, 8732, 10556, 19860, 29121, 32804, 4666, 2313, 27398, 14280, 2013, 29022, 26131, 21430, 21996, 95774, 49363, 12648, 54308, 6737, 8745, 11121, 49627

Offset: 1

Author

Jeppe Stig Nielsen, Apr 24 2022

Keywords

• nonn

Comments

The expression is the cyclotomic polynomial value Phi_{p^2}(b) where p=prime(n).

By definition, a(n) > 1. The occurrences of a(n)=2 correspond exactly to the terms of A156585.

Does a(n) tend to infinity (is liminf a(n) infinite)?

If it exists, a(27) > 857. - J.W.L. (Jan) Eerland, Dec 23 2022

a(65) = 1624. - Serge Batalov, Nov 17 2023

Links

• MersenneForum, Primes of the form (b^p^n-1)/(b^p^(n-1)-1) search thread

Crossrefs

Cf. A001248, A066180, A085398, A156585.

Programs

Mathematica

Table[k=2;Monitor[Parallelize[While[True,If[PrimeQ[(k^(Prime[n]^2)-1)/(k^Prime[n]-1)],Break[]];k++];k],k],{n,1,10}] (* J.W.L. (Jan) Eerland, Dec 22 2022 *)

PARI

forprime(p=2,,for(b=2,+oo,if(ispseudoprime(polcyclo(p^2,b)),print1(b,", ");break())))

Python

from sympy import isprime, prime
def a(n, startb=2):
    pn = prime(n); pn2 = pn**2; b = startb
    while not isprime((b**pn2-1)//(b**pn-1)): b += 1
    return b
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jun 21 2022

Formula

a(n) = A085398(prime(n)^2) = A085398(A001248(n)).

Extensions

a(25)-a(26) from J.W.L. (Jan) Eerland, Dec 23 2022

a(27) from Michael S. Branicky, Apr 04 2023

a(28)-a(33) from Martin Hopf, Nov 10 2023

a(34)-a(60) from Ryan Propper, Nov 17 2023