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A245512 Records in A245511: smallest m > 1 such that the largest odd number less than m^k is prime for every 0 < k < n, but not for k = n.

%I A245512 #31 Sep 18 2022 16:55:14
%S A245512 2,4,5,9,279,15331,1685775,205670529,129734299239,148778622108171
%N A245512 Records in A245511: smallest m > 1 such that the largest odd number less than m^k is prime for every 0 < k < n, but not for k = n.
%C A245512 For more comments and a program, see A245511. a(9), if it exists, certainly exceeds 500000000. It is not clear whether this sequence is infinite, nor whether a(n) is defined for every n.
%C A245512 For n > 2, a(n) is always odd, because A245511(i) can exceed 2 only when i is odd. Therefore to find more terms, it suffices to find odd bases m such that m-2, m^2-2, m^3-2, m^4-2, ..., m^N-2 is a long list of primes. - _Jeppe Stig Nielsen_, Sep 14 2022
%e A245512 a(3) = 5 because the odd numbers preceding 5^k, for k = 1,2,3, are 3, 23 and 123, and the first one which is not a prime corresponds to k = 3. Moreover, 5 is the smallest natural having this property.
%t A245512 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 < 210000000, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t (* _Robert G. Wilson v_, Aug 04 2014 *)
%o A245512 (PARI)
%o A245512 a(n) = for(k=1,10^6,c=0;for(i=1,n-1,if(isprime(k^i-(k%2)-1),c++));if(c==n-1&&!isprime(k^n-(k%2)-1),return(k)))
%o A245512 n=1;while(n<10,print1(a(n),", ");n++) \\ _Derek Orr_, Jul 27 2014
%Y A245512 Cf. A245509, A245510, A245511, A245513, A245514.
%K A245512 nonn,hard,more
%O A245512 1,1
%A A245512 _Stanislav Sykora_, Jul 24 2014
%E A245512 a(4) corrected by _Derek Orr_, Jul 27 2014
%E A245512 a(8) from _Robert G. Wilson v_, Aug 04 2014
%E A245512 a(9) from _Kellen Shenton_, Sep 13 2022
%E A245512 a(10) from _Kellen Shenton_, Sep 18 2022