[seqfan] Balanced Sophie Germain primes

Zak Seidov zakseidov at mail.ru
Sat Sep 17 20:00:27 CEST 2016

```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)
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
```