A066180 -id:A066180 - cp's OEIS frontend

A252503 Smallest prime p such that Phi_n(p) is also prime, where Phi is the cyclotomic polynomial, or 0 if no such p exists.

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 7, 2, 11, 3, 2, 113, 2, 43, 2, 2, 5, 151, 2, 2, 2, 2, 2, 179, 3, 61, 2, 23, 2, 53, 2, 89, 137, 11, 2, 5, 5, 2, 7, 73, 11, 307, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 0, 2, 53, 491, 197, 2, 3, 3, 3, 11, 19, 59, 7, 2, 2, 271, 2, 191, 61, 41, 7, 2, 2, 59, 5, 2, 2

Offset: 1

Eric Chen, Dec 18 2014

nonn

Conjecture: a(n) > 0 for all n != 2^k (k>5).
Clearly, if n is a power of 2, and Phi_n(2) is not prime, then a(n) = 0.
Records: 3, 5, 7, 11, 113, 151, 179, 307, 491, 839, 1427, 2411, 5987, 6389, 8933, 11813, 18587, 31721, 40763, 46349, ..., . - Robert G. Wilson v, May 21 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..1300 (first 1000 terms from Charles R Greathouse IV)

Cf. A123487, A123627, A085398, A066180, A103795.

Do[n=1; p=Prime[n]; cp=Cyclotomic[k, p]; While[!PrimeQ[cp], n=n+1; p=Prime[n]; cp=Cyclotomic[k, p]]; Print[p], {k, 1, 300}]

a(n)=if(n>>valuation(n,2)==1 && n>32, if(ispseudoprime(2^(n/2)+1), 2, 0), my(P=polcyclo(n)); forprime(p=2,, if(ispseudoprime(subst(P,'x,p)), return(p)))) \\ Charles R Greathouse IV, Dec 18 2014

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

Original entry on oeis.org

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

Jeppe Stig Nielsen, Apr 24 2022

nonn

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

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

Cf. A001248, A066180, A085398, A156585.

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 *)

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

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

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

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

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A252503 Smallest prime p such that Phi_n(p) is also prime, where Phi is the cyclotomic polynomial, or 0 if no such p exists.

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