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.

2, 4, 5, 9, 279, 15331, 1685775, 205670529, 129734299239, 148778622108171

Offset: 1

Author

Stanislav Sykora, Jul 24 2014

Keywords

• nonn
• hard
• more

Comments

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.

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

Examples

			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.
		

Crossrefs

Cf. A245509, A245510, A245511, A245513, A245514.

Programs

Mathematica

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 *)

PARI

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)))
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Jul 27 2014

Extensions

a(4) corrected by Derek Orr, Jul 27 2014

a(8) from Robert G. Wilson v, Aug 04 2014

a(9) from Kellen Shenton, Sep 13 2022

a(10) from Kellen Shenton, Sep 18 2022