A023221  Primes p such that 6*p + 5 is also prime

Similar to other prime sounds, this one is quite noisy. 

For terms of the sequence less than 10^6, the number
that fall into each residue class modulo m are as follows:

2   [1, 19096]
3   [1, 9509, 9587]
4   [0, 9514, 1, 9582]
5   [0, 4818, 4761, 4786, 4732]
6   [0, 9509, 1, 1, 0, 9586]
7   [1, 3897, 3823, 3795, 3716, 0, 3865]
8   [0, 4812, 1, 4762, 0, 4702, 0, 4820]
9   [0, 3139, 3251, 1, 3179, 3145, 0, 3191, 3191]
10   [0, 4818, 1, 4786, 0, 0, 0, 4760, 0, 4732]
11   [1, 0, 2104, 2124, 2147, 2126, 2145, 2165, 2101, 2077, 2107]
12   [0, 4766, 1, 1, 0, 4748, 0, 4743, 0, 0, 0, 4838]
13   [1, 1731, 1717, 1702, 1774, 1730, 1762, 1761, 1718, 1771, 0, 1712, 1718]
14   [0, 3897, 1, 3795, 0, 0, 0, 1, 0, 3822, 0, 3716, 0, 3865]

In the spectrogram the strongest spectral lines are at around 3150 (44100/14) Hz, 7350 (44100/6) Hz, and 14700 (44100/3) Hz. This may be related to the fact that the most non-uniformity at moduli of 3, 6, and 14.

An interesting pattern to note is that starting at mod 3, every other moduli only has one residue at 3 mod(n). I.E. there is one residue at 3 mod(4), 3 mod(6), 3 mod(8), etc.

--Kai vanNood

There are no elements in the sequence congruent to 5 modulo 7, since if p=5(mod 7),
then (6p+5)=0 (mod 7) and so is not prime (since 6p+5>7).

Similarly, there are no elements in the sequence congruent to 1 (mod 11), since if p = 1(mod 11),
then 6p+5 is divisible by 11, and so is not prime.