A350118 Primes p for which the smallest m such that p*2^m + 1 is prime increases. Sequence terminates with the smallest prime Sierpiński number.

2, 3, 7, 17, 19, 31, 47, 383, 2897, 3061, 5297, 7013, 10223

Offset

1,1

Comments

The smallest prime Sierpiński number is likely to be 271129.

Related to A058887: this sequence is A058887 with repeated values removed. The following list shows that relation between these two sequences:

a(2) = 3, A350119(2) = 1 => A058887(0..0) = 3;

a(3) = 7, A350119(3) = 2 => A058887(1..1) = 7;

a(4) = 17, A350119(4) = 3 => A058887(2..2) = 17;

a(5) = 19, A350119(5) = 6 => A058887(3..5) = 19;

a(6) = 31, A350119(6) = 8 => A058887(6..7) = 31;

a(7) = 47, A350119(7) = 583 => A058887(8..582) = 47;

a(8) = 383, A350119(8) = 6393 => A058887(583..6392) = 383;

...

a(N) is the smallest prime Sierpiński number, A350119(N) = -1 => A058887(k) = a(N) for all k >= A350119(N-1).

Links

Table of n, a(n) for n=1..13.

Example

Let b(p) be the smallest m such that p*2^m + 1 is prime. We have a(1) = 2 with b(2) = 0.
The least prime p such that b(p) > 0 is p = 3 with b(3) = 1, so a(2) = 3.
The least prime p such that b(p) > 1 is p = 7 with b(7) = 2, so a(3) = 7.
The least prime p such that b(p) > 2 is p = 17 with b(17) = 3, so a(4) = 17.
The least prime p such that b(p) > 3 is p = 19 with b(19) = 6, so a(5) = 19.
The least prime p such that b(p) > 6 is p = 31 with b(31) = 8, so a(6) = 31.
The least prime p such that b(p) > 8 is p = 47 with b(47) = 583, so a(7) = 47.

Prog

(PARI) b(p) = for(k=0, oo, if(isprime(p*2^k+1), return(k)))
list(lim) = if(lim>=2, my(v=[2], r=0); forprime(p=2, lim, if(b(p)>r, r=b(p); v=concat(v, p))); v)

Crossrefs

Cf. A058887, A057192, A350119, A064699, A076336 (Sierpiński numbers).

Sequence in context: A174359 A160513 A154431 * A256917 A089144 A171430

Adjacent sequences: A350115 A350116 A350117 * A350119 A350120 A350121

Keyword

nonn,fini,hard,more

Author

Jianing Song, Dec 14 2021

Status

approved