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%I #32 Jul 23 2019 16:53:12
%S 3,2,2,2,2,3,2,7,56,2,2,8,8,8,2,4,4,2,2,2,9,3,21496,26,2,2,4,38,7,
%T 286644,2,2,26,2,2,4,4,15,4,24,16,2,264,4,2,3,24,3,516,6
%N Least k>1 such that p^k - p^(k-1) - 1 is prime for p = prime(n).
%C Does a(n) always exist? Note that k cannot be 5, 11, 17,... (i.e., k=5 mod 6) because then p^2 - p + 1 divides p^k - p^(k-1) - 1.
%C From _Richard N. Smith_, Jul 15 2019: (Start)
%C The link has the primes 82*83^21495-1 = 83^21496-83^21495-1 and 112*113^286643-1 = 113^286644-113^286643-1, thus a(23)=21496 and a(30)=286644.
%C a(51) > 250000, since 232*233^k-1 is composite for all k<=250000, see link.
%C a(52) - a(61) = {4, 2, 80, 14, 76, 2, 90, 6, 80, 769}, a(62) > 200000. (End)
%H Steven Harvey, <a href="http://harvey563.tripod.com/wills.txt">Williams primes</a>
%t lst={}; Do[p=Prime[n]; k=2; While[m=p^k-p^(k-1)-1; !PrimeQ[m], k++ ]; AppendTo[lst,k], {n,22}]; lst
%o (PARI) a(n)=for(k=2, 10^6, if(ispseudoprime(prime(n)^k - prime(n)^(k-1) - 1), return(k))) \\ _Richard N. Smith_, Jul 15 2019
%Y Cf. A087139, A122395.
%K nonn,more,hard
%O 1,1
%A _T. D. Noe_, Aug 31 2006
%E a(23)-a(50) from _Richard N. Smith_, Jul 15 2019, using Steven Harvey's table.
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