A051860 Distinct prime numbers in order of their appearance in A051686.

2, 3, 7, 5, 19, 13, 37, 11, 31, 29, 61, 17, 23, 103, 83, 41, 47, 43, 107, 53, 97, 59, 67, 79, 73, 71, 109, 167, 151, 197, 127, 163, 89, 137, 157, 139, 113, 233, 181, 101, 229, 193, 179, 211, 131, 149, 373, 251, 271, 307, 173, 409, 199, 283, 227, 349, 263, 443, 313

Offset: 1

Author

Labos Elemer, Dec 14 1999

Keywords

• nonn

Comments

A051686 includes the minimal 2k-Germain primes, which show a special order of emergence as 2k increases. This sequence shows this order.

Each prime seems to appear several times.

Examples

			37 appears first in A051686 at the 148th position and it is the 7th new prime number which arises, sooner than smaller primes like 11, 31, 29, 17, 23, so a(7) = 37.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A051686, A005384, A051692.

Programs

Mathematica

s[n_] := Module[{p = 2}, While[! PrimeQ[2*n*p + 1], p = NextPrime[p]]; p]; seq[len_] := Module[{t = {}, n = 1, c = 0, p}, While[c < len, p = s[n]; If[FreeQ[t, p], c++; AppendTo[t, p]]; n++]; t]; seq[60] (* Amiram Eldar, Feb 28 2025 *)

PARI

s(n) = {my(p = 2); while(!isprime(2*n*p + 1), p = nextprime(p+1)); p;}
isin(list, k) = {for(i = 1, #list, if(list[i] == k, return(1))); 0};
list(len) = {my(t = List(), n = 1, c = 0, p); while(c < len, p = s(n); if(!isin(t, p), c++; listput(t, p)); n++); Vec(t);} \\ Amiram Eldar, Feb 28 2025