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%I #22 Feb 17 2020 07:04:42
%S 1,1,1,1,5,1,1,1,5,2,1,39,6,4,12,2,2,1,6,17,46,7,5,1,25,2,41,1,12,7,1,
%T 7,327,7,8,44,26,12,75,14,51,110,4,14,49,286,15,4,39,22,109,367,22,67,
%U 27,95,80,149,2,142,3,11,402,3,44,10,82,20,95,4,108,349,127,303,37,3,162
%N Number k yielding the smallest prime of the form (k+1)^p - k^p, where p = prime(n).
%C The smallest k generating a prime of the form (k+1)^p - k^p (A121620) for the prime A000040(n). For the primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... (A000043), k = 1 and Mersenne primes 2^p - 1 (A000668) are obtained. For p = 11, 23, 29, ..., the smallest primes of the form (k+1)^p - k^p are respectively 313968931 (for k = 5), 777809294098524691 (for k = 5 also), 68629840493971 (for k = 2), ..., so a(5) = 5, a(9) = 5, a(10) = 2, ...
%H Jinyuan Wang, <a href="/A222119/b222119.txt">Table of n, a(n) for n = 1..175</a>
%F a(n) = A103794(n) - 1. - _Ray Chandler_, Feb 26 2017
%p A222119 := proc(n)
%p p := ithprime(n) ;
%p for k from 1 do
%p if isprime((k+1)^p-k^p) then
%p return k;
%p end if;
%p end do:
%p end proc: # _R. J. Mathar_, Feb 10 2013
%t Table[p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; k, {n, 80}] (* _T. D. Noe_, Feb 12 2013 *)
%o (PARI) f(p) = {my(k=1); while(ispseudoprime((k+1)^p-k^p)==0, k++); k; }
%o lista(nn) = forprime(p=2, nn, print1(f(p), ", ")); \\ _Jinyuan Wang_, Feb 03 2020
%Y Cf. A103794, A222120 (number of digits in the primes).
%K nonn
%O 1,5
%A _Vladimir Pletser_, Feb 07 2013
%E More terms from _Ray Chandler_, Feb 27 2017
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