%I #11 Dec 23 2024 14:53:44
%S 2,2,29,3299,4866623,22081407211439
%N First of n consecutive Sophie Germain primes (A005384: such that 2p+1 is also prime) in arithmetic progression.
%C The corresponding safe primes 2p+1 (A005385) are again the first in that sequence to have the same property.
%C Terms a(5) and a(6) were given, respectively, by Neil Fernandez and Giovanni Resta, on the SeqFan mailing list, cf. links.
%H Giovanni Resta, in reply to Harvey P. Dale and others, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-September/016776.html">Re: Consecutive Sophie Germain primes with the same gap</a>, SeqFan mailing list, Sep. 2016. (Click "Previous message" to see <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-September/016771.html">Neil Fernandez' earlier results</a>.)
%e 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.
%e 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.
%e 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.
%e 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.
%e 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).
%Y Cf. A005384 (Sophie Germain primes), A074259 (gaps between SG primes), A005385 (safe primes: 2p+1 for SG primes p).
%K nonn
%O 1,1
%A _M. F. Hasler_, Sep 18 2016