A293559 - OEIS

A293559

Least prime p such that n*(p^n-1)-1 is prime.

5, 2, 11, 2, 5, 29, 7, 2, 47, 71, 167, 2, 571, 23, 59, 2, 73, 53, 349, 5, 59, 1259, 769, 17, 1021, 1117, 73, 5, 1049, 5, 109, 137, 947, 29, 89, 1019, 67, 29, 97, 2, 2111, 569, 271, 53, 191, 5, 251, 113, 2029, 569, 17, 1453, 1049, 1151, 211, 7, 47, 677, 29

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Note that f_n(p) = n*(p^n-1)-1 = n*p^n-(n+1) is irreducible over the rationals as a polynomial in p: if n <> 8 Eisenstein's criterion applies to either f_n(p) or its reversal -(n+1)*p^n+n, using Mihailescu's theorem. Thus the generalized Bunyakovsky conjecture implies a(n) always exists. - Robert Israel, Oct 24 2017

LINKS

Iain Fox, Table of n, a(n) for n = 1..1000

EXAMPLE

For n=5, 5*(5^5-1)-1 = 15619 is prime, but 5*(p^5-1)-1 is not prime for primes p < 5, so a(5)=5.

MAPLE

f:= proc(n) local p;

p:= 2;

while not isprime(n*(p^n-1)-1) do p:= nextprime(p) od:

end proc:

map(f, [$1..100]); # Robert Israel, Oct 24 2017

MATHEMATICA

Table[p=2; While[!PrimeQ[n (p^n - 1) - 1], p=NextPrime@p]; p, {n, 100}]