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Smallest number b such that b^prime(n) - (b-1)^prime(n) is prime.
4

%I #15 Jun 04 2024 17:05:42

%S 2,2,2,2,6,2,2,2,6,3,2,40,7,5,13,3,3,2,7,18,47,8,6,2,26,3,42,2,13,8,2,

%T 8,328,8,9,45,27,13,76,15,52,111,5,15,50,287,16,5,40,23,110,368,23,68,

%U 28,96,81,150,3,143,4,12,403,4,45,11,83,21,96,5,109,350,128,304,38,4,163

%N Smallest number b such that b^prime(n) - (b-1)^prime(n) is prime.

%C Conjecture: sequence is defined for all positive indices.

%C For p=prime(n), Eisenstein's irreducibility criterion can be used to show that the polynomial (x+1)^p-x^p is irreducible, which is a necessary (but not sufficient) condition for a(n) to exist. - _T. D. Noe_, Dec 05 2005

%F a(n) = A222119(n) + 1. - _Ray Chandler_, Feb 26 2017

%e 2^prime(1)-1^prime(1)=3 is prime, so a(1)=2;

%e 2^prime(5)-1^prime(5)=2047 has a factor of 23;

%e ...

%e 6^prime(5)-5^prime(5)=313968931 is prime, so a(5)=6;

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

%p p:= ithprime(n);

%p for b from 2 do

%p if isprime(b^p - (b-1)^p) then return b fi

%p od

%p end proc:

%p map(f, [$1..80]); # _Robert Israel_, Jun 04 2024

%t Do[p=Prime[k]; n=2; nm1=n-1; cp=n^p-nm1^p; While[ !PrimeQ[cp], n=n+1; nm1=n-1; cp=n^p-nm1^p]; Print[n], {k, 1, 200}]

%Y Cf. A103795, A066180, A058013, A222119.

%K nonn

%O 1,1

%A _Lei Zhou_, Feb 24 2005