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A353101
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Least b > 1 such that (b^(prime(n)^2) - 1)/(b^prime(n) - 1) is prime.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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)?
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(28)-a(33) from Martin Hopf, Nov 10 2023
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STATUS
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approved
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