A339474 Numbers at the start of a run of 2 or more consecutive primes that are Sophie Germain primes.

2, 23, 83, 173, 233, 653, 1013, 1223, 1499, 1889, 2063, 2393, 2543, 2693, 2963, 3803, 4373, 5039, 6101, 6263, 6323, 6491, 7079, 7643, 7883, 9473, 10691, 13883, 14153, 14303, 15161, 16811, 17669, 19553, 19913, 20753, 21701, 22259, 22343, 22409, 22469, 25913

Offset

1,1

Comments

This sequence is similar to A128825, but also includes 2, and omits numbers like 20759 (see Example), which is a Sophie Germain prime in the interior of a run of more than two consecutive primes that are Sophie Germain primes.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

2 is a Sophie Germain prime (since 2*2 + 1 = 5 is prime), 3 is a Sophie Germain prime (since 2*3 + 1 = 7 is prime), and 2 and 3 are consecutive primes, so 2 is a term of the sequence.
23 and 29 are consecutive primes, and both 23*2 + 1 = 47 and 29*2 + 1 = 59 are primes, so 23 is a term.
The three consecutive primes 20753, 20759, and 20771 are all Sophie Germain primes, and the primes immediately before 20753 and immediately after 20771 are not Sophie Germain primes, so {20753, 20759, 20771} forms a run of three consecutive primes that are Sophie Germain primes; 20753 is at the start of the run, so it is a term, but 20759 is not at the start of the run, so it is not a term.

Maple

P:= [seq(ithprime(i), i=1..10000)]:
SG:= map(t -> isprime(2*t+1), P):
P[select(j -> j=1 or SG[j-1..j+1]=[false, true, true], [$1..9999])]; # Robert Israel, Apr 28 2021

Mathematica

Select[Prime@Range@3000, And@@PrimeQ[{2#+1, 2NextPrime@#+1}]&] (* Giorgos Kalogeropoulos, Apr 28 2021 *)

Crossrefs

Cf. A005384 (Sophie Germain primes), A128825, A338700, A339475, A339476, A339477, A339478.

Sequence in context: A345701 A209194 A097232 * A222564 A099134 A069152

Adjacent sequences: A339471 A339472 A339473 * A339475 A339476 A339477

Keyword

nonn

Author

Jon E. Schoenfield, Apr 24 2021

Status

approved