A276821 First of n consecutive Sophie Germain primes (A005384: such that 2p+1 is also prime) in arithmetic progression.

2, 2, 29, 3299, 4866623, 22081407211439

Offset: 1

Author

M. F. Hasler, Sep 18 2016

Keywords

• nonn

Comments

The corresponding safe primes 2p+1 (A005385) are again the first in that sequence to have the same property.

Terms a(5) and a(6) were given, respectively, by Neil Fernandez and Giovanni Resta, on the SeqFan mailing list, cf. links.

Examples

			The first two consecutive identical gaps between Sophie Germain primes are 12 and 12 which occur between A005384(6..8) = (29, 41, 53), therefore a(3) = 29.
The first three consecutive identical gaps between Sophie Germain primes are equal to 30 and occur between A005384(85..88) = (3299, 3329, 3359, 3389), therefore a(4) = 3299.
The first four consecutive identical gaps between Sophie Germain primes are equal to 150 and occur between A005384(29952..29956) = (4866623, 4866773, 4866923, 4867073, 4867223), therefore a(5) = 4866623.
The first five consecutive identical gaps between Sophie Germain primes are equal to 420 and occur between A005384(32361449747..32361449752) = (22081407211439, 22081407211859, 22081407212279, 22081407212699, 22081407213119, 22081407213539), therefore a(6) = 22081407211439.
For n=1 and n=2, a(n) is equal to the smallest Sophie Germain prime, A005384(1) = 2, which is the first of two terms (and also one term) "in arithmetic progression" (which means not any restriction for a single term or any two subsequent terms).
		

Links

• Giovanni Resta, in reply to Harvey P. Dale and others, Re: Consecutive Sophie Germain primes with the same gap, SeqFan mailing list, Sep. 2016. (Click "Previous message" to see Neil Fernandez' earlier results.)

Crossrefs

Cf. A005384 (Sophie Germain primes), A074259 (gaps between SG primes), A005385 (safe primes: 2p+1 for SG primes p).