A005658: If n appears so do 2n, 3n+2, 6n+3.

Noisy static. Fluctuates rapidly, but occurs consistently throughout the sound. 

Residue counts:
2   [259928, 96527]
3   [129448, 76725, 150282]
4   [156517, 26460, 103411, 70067]
5   [71302, 71203, 71321, 71324, 71305]
6   [66363, 1, 116841, 63085, 76724, 33441]
7   [50778, 51165, 51190, 50813, 50885, 50898, 50726]
8   [85638, 10385, 43045, 41960, 70879, 16075, 60366, 28107]
9   [27717, 20335, 53254, 49750, 27289, 39880, 51981, 29101, 57148]
10   [52027, 19262, 51996, 19347, 51987, 19275, 51941, 19325, 51977, 19318]
11   [32386, 32499, 32425, 32382, 32324, 32380, 32437, 32433, 32395, 32413, 32381]
12   [34049, 1, 53878, 45866, 59505, 9240, 32314, 0, 62963, 17219, 17219, 24201]

Density information:
A(x) at x=10,100,1000,...:
[7, 52, 460, 4205, 38400, 356455]

A(x)/x at x=10,100,1000,...:
0.7,0.52,0.46,0.4205,0.384,0.356455,

First 100 first differences:
[1, 2, 1, 3, 1, 1, 4, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 2, 1, 1, 2, 4, 4, 3, 3, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 4, 8, 3, 3, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 4, 2, 1, 1, 2, 2, 4, 2, 2, 7, 1, 8, 3, 3, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2]

First 99 second differences:
[1,-1,2,-2,0,3,-3,0,0,0,1,4,-5,0,0,0,1,-1,0,1,2,0,-1,0,-2,0,0,0,1,-1,0,1,0,-1,0,1,0,2,4,-5,0,-2,0,0,2,-2,0,3,-3,0,0,0,1,-1,0,1,0,-1,0,1,0,2,-2,-1,0,1,0,2,-2,0,5,-6,7,-5,0,-2,0,0,2,-2,0,3,-3,0,0,0,1,-1,0,3,-3,0,1,0,1,0,-2,0,1]

This is a quite dense sequence with 356455 terms less than 10^6. From the spectrogram, there are distinct lines at the approximate frequencies: 3671Hz, 7348Hz, 11012Hz, and 14689Hz. The following frequencies are close to the multiples of 3671Hz (x2, x3, x4). Dividing the sampling rate by these frequencies, we get 44100/12, 44100/6, 44100/4, and 44100/3. For mod 3, there is some non-uniformity in the distribution. 2 mod 3 occurs most often and 1 mod 3 occurs least often. If n is in the sequence, then 3n + 2 is always congruent to 2 mod 3, while 6n + 3 is always congruent to 0 mod 3. Therefore, 1 mod 3 is less common because it mainly appears through the 2n rule when n is congruent to 2 mod 3. For mod 4, the distribution is also not uniform. 0 mod 4 has the most terms, while 1 mod 4 has far fewer. This makes sense because the rule 2n always produces an even number, so it can only contribute to 0 or 2 mod 4. Thus, 1 mod 4 is not directly produced by the 2n rule and appears less often. Next, mod 6's distribution is fairly non-uniform. 1 mod 6 appears only once from the starting term 1, while 2 mod 6 has the most terms. The rule 2n can only produce even residue classes mod 6 (0, 2, 4). 3n + 2 produces only 2 or 5 mod 6, while 6n + 3 is always congruent to 3 mod 6. We can see that 2 mod 6 has the most terms because it can be generated by both 2n and 3n + 2. Lastly, for mod 12, 7 mod 12 has 0 terms and 1 mod 12 has only 1 term. This is due to 2n always producing an even number, so it can only contribute to even residue classes mod 12 (0, 2, 4, 6, 8, 10). The rule 3n + 2 can only produce 2, 5, 8, or 11 mod 12, depending on n. 6n + 3 can only produce 3 or 9 mod 12. Therefore, 1 mod 12 only appears once because it comes from the starting value 1, and 7 mod 12 never appears because none of the rules produce it.

The sequence is infinite because if we focus only on the rule 2n, we can generate new terms forever. Since 1 is in the sequence, repeatedly applying this rule gives 2, 4, 8, 16, and so on. The sequence contains the infinite subsequence 1, 2, 4, 8, 16,..., which shows that the full sequence is infinite.

-- Lauren Yee