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Least k such that k^prime(n) - k^((prime(n)+1)/2) + 1 is prime for n > 1.
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%I #17 May 07 2024 01:59:30

%S 2,5,2,3,5,5,9,26,9,16,11,5,6,2,46,18,16,89,10,2,2,94,7,7,16,26,230,5,

%T 2,140,34,69,114,6,2,179,994,2,76,165,8,69,3,294,230,96,7,720,684,

%U 2029,2,2,25,135,523,271,161,1210,139,14,34,194,238,87,355,636,40,1114,519,2

%N Least k such that k^prime(n) - k^((prime(n)+1)/2) + 1 is prime for n > 1.

%e 2^3 - 2^2 + 1 = 5, which is prime, so a(2) = 2.

%e 2^5 - 2^3 + 1 = 25 = 5*5, and

%e 3^5 - 3^3 + 1 = 217 = 7*31, but

%e 5^5 - 5^3 + 1 = 3001, which is prime, so a(3) = 5.

%t seq={};Do[k=1;Until[PrimeQ[k^Prime[n]-k^((Prime[n]+1)/2)+1],k++];AppendTo[seq,k],{n,2,71}];seq (* _James C. McMahon_, May 06 2024 *)

%o (PARI) a(n) = {my(k=1, p=prime(n)); while (!isprime(k^p-k^((p+1)/2)+1), k++); k;} \\ _Michel Marcus_, Sep 16 2019

%K nonn

%O 2,1

%A _Pierre CAMI_, Oct 28 2005

%E a(33) and a(34) concatenated by _Georg Fischer_, Jun 22 2022