# [seqfan] Re: Balanced Sophie Germain primes

Harvey P. Dale hpd at hpdale.org
Mon Sep 19 15:49:30 CEST 2016

```	When this sequence is submitted, the following Mma program can be added as it generates the terms of the sequence:

Select[Partition[Select[Prime[Range[20000]],PrimeQ[2#+1]&],3,1],Mean[#]==#[[2]]&][[All,2]]

Best,
Harvey

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of M. F. Hasler
Sent: Sunday, September 18, 2016 5:26 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Balanced Sophie Germain primes

On Sat, Sep 17, 2016 at 2:00 PM, Zak Seidov via SeqFan < seqfan at list.seqfan.eu> wrote:

> Balanced SG primes (of order one) : SG primes which are the average of
> the previous SG prime and the following SG prime ( = Sophie Germain
> prime
> A005384)
>

I think this is an interesting sequence because it yields (the second of) any subsequence of at least 3 terms (consecutive SG primes) in arithmetic progression, and so it may also serve to find longer subsequences of AP's within SG primes (namely when the distance to the next term is equal to the gap for the given triple, as is the case for 3329, 3359 which are at distance of 30, equal to "their" gap).
(Remarkably enough, {3329,3359} are also both the lesser of a twin prime pair and also have prime(p)+2 prime and appear as consecutive terms in a few other sequences <https://oeis.org/search?q=3329%2C+3359> !)
--
Maximilian

41, 431, 1811, 3329, 3359, 4391, 6521, 11549, 14081, 14741, 14831, 18191,
> 19991, 21803, 25673, 28001, 31721, 32933, 36791, 43691, 44189, 49481,
> 51521, 59021, 59981, 68669, 74729, 75041, 85223, 104759, 111641,
> 112571, 120671, 125201, 126683, 136463, 139721, 155009, 162251,
> 166781, 170003, 179603, 193841, 196541 Apparently the sequence is
> infinite (and the same is apparently true for balanced primes of
> higher order) Corresponding gaps: 12, 12, 78, 30, 30, 18, 30, 30, 72,
> 42, 48, 42, 72, 90, 30, 18, 72, 90, 30, 30, 60, 18, 18, 42, 102, 30,
> 30, 108, 90, 30, 18, 12, 108, 198, 30, 60, 18, 210, 42, 42, 60, 30,
> 30, 102 (all multiples of 6).
>      Note that corresponding safe primes are also balanced safe primes.
>      Some 10000-strong  bfile and comments are welcome.
>  Cf. A006562 Balanced primes (of order one) : primes which are the
> average of the previous prime and the following prime.
> Cf. A005384 Sophie Germain primes.
> Cf. A005385 Safe primes p:(p-1)/2 is also prime.
> seq--
> Zak  Seidov
>

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