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A293559 Least prime p such that n*(p^n-1)-1 is prime. 1

%I

%S 5,2,11,2,5,29,7,2,47,71,167,2,571,23,59,2,73,53,349,5,59,1259,769,17,

%T 1021,1117,73,5,1049,5,109,137,947,29,89,1019,67,29,97,2,2111,569,271,

%U 53,191,5,251,113,2029,569,17,1453,1049,1151,211,7,47,677,29

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

%C 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

%H Iain Fox, <a href="/A293559/b293559.txt">Table of n, a(n) for n = 1..1000</a>

%e 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.

%p f:= proc(n) local p;

%p p:= 2;

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

%p p

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 24 2017

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

%o (PARI) a(n) = forprime(p=2, , if(ispseudoprime(n*(p^n-1)-1), return(p))) \\ _Iain Fox_, Oct 23 2017

%Y Cf. A283450.

%K nonn

%O 1,1

%A _Vincenzo Librandi_, Oct 12 2017

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Last modified January 17 22:33 EST 2022. Contains 350410 sequences. (Running on oeis4.)