A006562: Balanced primes (of order one): primes which are the average of the previous prime and the following prime. 

Crunchy static noise.

Residue counts:
2   [0, 2994]
3   [0, 1481, 1513]
4   [0, 1512, 0, 1482]
5   [1, 507, 989, 1036, 461]
6   [0, 1481, 0, 0, 0, 1513]
7   [0, 264, 575, 672, 679, 541, 263]
8   [0, 758, 0, 736, 0, 754, 0, 746]
9   [0, 495, 507, 0, 475, 522, 0, 511, 484]
10   [0, 507, 0, 1036, 0, 1, 0, 989, 0, 461]
11   [0, 299, 375, 350, 371, 141, 116, 366, 349, 362, 265]
12   [0, 740, 0, 0, 0, 772, 0, 741, 0, 0, 0, 741]

First 100 first differences:
[48, 104, 16, 38, 46, 6, 110, 190, 30, 14, 46, 80, 214, 30, 126, 20, 64, 36, 144, 144, 236, 6, 154, 380, 130, 260, 226, 60, 344, 6, 324, 96, 280, 396, 48, 140, 60, 34, 302, 114, 6, 190, 84, 6, 170, 244, 266, 190, 54, 6, 44, 6, 490, 114, 546, 60, 240, 18, 276, 596, 34, 650, 76, 398, 382, 354, 50, 196, 446, 112, 86, 234, 76, 96, 30, 564, 50, 36, 64, 6, 188, 132, 70, 134, 280, 6, 1080, 204, 6, 408, 32, 120, 154, 300, 30, 6, 104, 6, 184, 126]

The sequence has 2994 terms less than 10^6, which is highly sparse and gives the sound a crunchy static texture. Noticeably, the spectrogram has a few white streaks and no distinct spectral lines. Upon looking at the plot spectrum, there are interesting triangle shapes or peaks with wide bases centered at approximately 7324Hz and 14699Hz. The frequency 14699Hz is close to a multiple of 7324Hz, and the plot spectrum also appears roughly symmetric around 11007Hz, which is close to 44100/4. Dividing the sampling rate by 7324Hz and 14699Hz gives 44100/6 and 44100/3. One thing to notice about mod 3 and mod 6 is that their residue counts are directly connected. For mod 3, residue 1 has 1481 terms and residue 2 has 1513 terms. For mod 6, residue 1 also has 1481 terms and residue 5 has 1513 terms. This makes sense because any number congruent to residue 1 mod 6 is congruent to residue 1 mod 3, while any number congruent to residue 5 mod 6 is congruent to residue 2 mod 3 because 3 is a divisor of 6. Also, residue 0 mod 3 has 0 terms because primes greater than 3 cannot be divisible by 3. Primes greater than 3 must also be congruent to either 1 or 5 mod 6, which is why only those two residue classes contain terms. The zeros in the residue counts come from the fact that every term is prime. All balanced primes greater than 3 are odd, so even residue classes have 0 terms in mod 2, mod 4, mod 8, mod 10, and mod 12.

It is currently unknown whether this sequence is infinite. In modern number theory, it has not been proven that there are infinitely many balanced primes.

-- Lauren Yee