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

1,1

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

Links

Table of n, a(n) for n=1..10.

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A067298 A077389 A122991 * A366351 A378098 A232616

Adjacent sequences: A245509 A245510 A245511 * A245513 A245514 A245515

Keyword

nonn,hard,more

Author

Stanislav Sykora, Jul 24 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

Status

approved