A051692 -id:A051692 - cp's OEIS frontend

A051861 Twice the positions in A051686 at which new primes appear in that sequence.

2, 4, 16, 38, 58, 72, 148, 170, 178, 282, 388, 446, 512, 652, 656, 758, 836, 856, 1514, 1592, 1648, 1712, 1906, 1918, 2034, 2606, 2758, 4094, 4936, 5624, 5758, 5842, 5924, 6764, 8188, 10570, 10922, 11072, 11482, 13082, 13972, 14626, 15212, 16018, 18986

Offset: 1

Labos Elemer, Dec 14 1999

nonn

Halving this sequence gives 1, 2, 8, 19, ..., 256, 326, ..., 47764, ..., the indices in A051686 at which primes appear which have not appeared before.

This sequence lists the terms of A051692 in ascending order, whereas A051692 lists them in increasing order of the emerging primes in A051686.

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

Cf. A051686, A051692, A005384.

s[n_] := Module[{p = 2}, While[! PrimeQ[2*n*p + 1], p = NextPrime[p]]; p]; seq[len_] := Module[{t = {}, v = {}, n = 1, c = 0, p}, While[c < len, p = s[n]; If[FreeQ[t, p], c++; AppendTo[t, p]; AppendTo[v, 2*n]]; n++]; v]; seq[45] (* Amiram Eldar, Feb 28 2025 *)

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); print1(2*n, ", ")); n++);} \\ Amiram Eldar, Feb 28 2025

Edited by Jon E. Schoenfield, May 28 2018

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

Original entry on oeis.org

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

Labos Elemer, Dec 14 1999

nonn

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.

			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.

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

Cf. A051686, A005384, A051692.

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 *)

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

cp's OEIS Frontend

A051861 Twice the positions in A051686 at which new primes appear in that sequence.

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