%I
%S 7,5,2,105,3,909,4995825,28212939
%N Records in A245509: smallest m > 1 such that the first odd number greater than m^k is prime for every 0 < k < n, but not for k = n.
%C For more comments and a program, see A245509. a(9), if it exists, certainly exceeds 1050000000. It is not clear whether this sequence is infinite, nor whether a(n) is defined for every n.
%e a(4) = 105 because 105 is the smallest m such that the first odd numbers after m^k are prime for k = 1,2,3, but composite for k = 4.
%t f[n_] := Block[{d = If[ OddQ@ n, 2, 1], m = 1, t}, While[t = n^m + d; EvenQ@ t  PrimeQ@ t, m++]; m]; t = Table[0, {25}]; k = 2; While[k < 29000000, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t (* _Robert G. Wilson v_, Aug 04 2014 *)
%o (PARI)
%o a(n) = for(k=1, 10^6, c=0; for(i=1, n1, if(isprime(k^i+(k%2)+1), c++)); if(c==n1&&!isprime(k^n+(k%2)+1), return(k)))
%o n=1;while(n<10,print1(a(n),", ");n++) \\ _Derek Orr_, Jul 27 2014
%Y Cf. A245509, A245511, A245512, A245513, A245514.
%K nonn,hard,more
%O 1,1
%A _Stanislav Sykora_, Jul 24 2014
%E a(4) and example corrected by _Derek Orr_, Jul 27 2014
%E a(8) from _Robert G. Wilson v_, Aug 04 2014
