A002081: Numbers congruent to {2, 4, 8, 16} (mod 20).

Constant high-pitched sound.

Residue counts:
2   [200000, 0]
3   [66666, 66668, 66666]
4   [150000, 0, 50000, 0]
5   [0, 50000, 50000, 50000, 50000]
6   [66666, 0, 66666, 0, 66668, 0]
7   [28572, 28572, 28571, 28570, 28572, 28571, 28572]
8   [75000, 0, 25000, 0, 75000, 0, 25000, 0]
9   [22221, 22223, 22222, 22222, 22223, 22221, 22223, 22222, 22223]
10   [0, 0, 50000, 0, 50000, 0, 50000, 0, 50000, 0]
11   [18183, 18181, 18183, 18181, 18182, 18182, 18181, 18182, 18182, 18182, 18181]
12   [50000, 0, 16667, 0, 50001, 0, 16666, 0, 49999, 0, 16667, 0]

Density information:
A(x) at x=10,100,1000,...:
[3, 20, 200, 2000, 20000, 200000]

A(x)/x at x=10,100,1000,...:
0.3,0.2,0.2,0.2,0.2,0.2,

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

First 99 second differences:
[2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2,-4,2,4,-2]

From the spectrogram, the majority of frequencies are faint, with numerous that appear as very dark spectral lines. The spectral lines represent the approximate frequencies: 2201Hz, 4417Hz, 6617Hz, 8822Hz, 11055Hz, 13226Hz, 15442Hz, 17642Hz, and 19847Hz. Noticeably, these frequencies are close to integer multiples of 2201Hz, which equals about 44100Hz/20. It makes sense that the frequencies are based on 20 because the sequence repeats the pattern {2, 4, 8 16}, but adds 20 to each new group. 

All the terms in the sequence are even since 2, 4, 8, and 16 are all even. This is reflected in mod 2. Also, the odd residue classes are 0 for mod 4, 6, 8, 10, and 12. For mod 4, only residues 0 and 2 appear. Residue 0 appears more often because 4, 8, and 16 are all divisible by 4, while 2 gives residue 2. For mod 5, residue 0 never appears because none of the repeating residues 2, 4, 8, 16 are divisible by 5. Instead, they become 2, 4, 3, 1 (mod 5), so residues 1 through 4 each appear evenly. The same logic can be applied with mod 10. Only residues 2, 4, 6, and 8 appear because 2, 4, 8, 16 become 2, 4, 8, 6 (mod 10). Further, the distributions of counts for mod 3, 7, 9, 11 are not exactly uniform because there are 200000 terms total, and the terms do not always divide evenly into each number of residue groups. 

The density is around 0.2 because out of every 20 numbers, only 4 of them match the pattern (numbers congruent to 2, 4, 8, or 16 mod 20). Since the sequence simply relies on the pattern of {2, 4, 8, 16} with 20 added each iteration, the sequence is infinite.

--Lauren Yee