A353101 - OEIS

%I #44 Nov 17 2023 17:51:45

%S 2,2,22,2,43,24,315,38,54,265,605,61,697,306,1153,370,2,10688,3075,

%T 2338,1153,3243,130,2301,315,200,1155,14739,4591,2230,263,6665,250,

%U 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

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

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

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

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

%C If it exists, a(27) > 857. - _J.W.L. (Jan) Eerland_, Dec 23 2022

%C a(65) = 1624. - _Serge Batalov_, Nov 17 2023

%H MersenneForum, <a href="https://mersenneforum.org/showthread.php?t=28998">Primes of the form (b^p^n-1)/(b^p^(n-1)-1) search thread</a>

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

%t 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 *)

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

%o (Python)

%o from sympy import isprime, prime

%o def a(n, startb=2):

%o     pn = prime(n); pn2 = pn**2; b = startb

%o     while not isprime((b**pn2-1)//(b**pn-1)): b += 1

%o     return b

%o print([a(n) for n in range(1, 10)]) # _Michael S. Branicky_, Jun 21 2022

%Y Cf. A001248, A066180, A085398, A156585.

%K nonn

%O 1,1

%A _Jeppe Stig Nielsen_, Apr 24 2022

%E a(25)-a(26) from _J.W.L. (Jan) Eerland_, Dec 23 2022

%E a(27) from _Michael S. Branicky_, Apr 04 2023

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

%E a(34)-a(60) from _Ryan Propper_, Nov 17 2023