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

%I A051860 #17 Feb 28 2025 05:56:59
%S A051860 2,3,7,5,19,13,37,11,31,29,61,17,23,103,83,41,47,43,107,53,97,59,67,
%T A051860 79,73,71,109,167,151,197,127,163,89,137,157,139,113,233,181,101,229,
%U A051860 193,179,211,131,149,373,251,271,307,173,409,199,283,227,349,263,443,313
%N A051860 Distinct prime numbers in order of their appearance in A051686.
%C A051860 A051686 includes the minimal 2k-Germain primes, which show a special order of emergence as 2k increases. This sequence shows this order.
%C A051860 Each prime seems to appear several times.
%H A051860 Amiram Eldar, <a href="/A051860/b051860.txt">Table of n, a(n) for n = 1..400</a>
%e A051860 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.
%t A051860 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 *)
%o A051860 (PARI) s(n) = {my(p = 2); while(!isprime(2*n*p + 1), p = nextprime(p+1)); p;}
%o A051860 isin(list, k) = {for(i = 1, #list, if(list[i] == k, return(1))); 0};
%o A051860 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
%Y A051860 Cf. A051686, A005384, A051692.
%K A051860 nonn
%O A051860 1,1
%A A051860 _Labos Elemer_, Dec 14 1999