A350118 - OEIS

%I #18 Dec 28 2021 04:18:14

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

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

%C The smallest prime Sierpiński number is likely to be 271129.

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

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

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

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

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

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

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

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

%C ...

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

%e Let b(p) be the smallest m such that p*2^m + 1 is prime. We have a(1) = 2 with b(2) = 0.

%e The least prime p such that b(p) > 0 is p = 3 with b(3) = 1, so a(2) = 3.

%e The least prime p such that b(p) > 1 is p = 7 with b(7) = 2, so a(3) = 7.

%e The least prime p such that b(p) > 2 is p = 17 with b(17) = 3, so a(4) = 17.

%e The least prime p such that b(p) > 3 is p = 19 with b(19) = 6, so a(5) = 19.

%e The least prime p such that b(p) > 6 is p = 31 with b(31) = 8, so a(6) = 31.

%e The least prime p such that b(p) > 8 is p = 47 with b(47) = 583, so a(7) = 47.

%o (PARI) b(p) = for(k=0, oo, if(isprime(p*2^k+1), return(k)))

%o 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)

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

%K nonn,fini,hard,more

%O 1,1

%A _Jianing Song_, Dec 14 2021