A005349: Niven (or Harshad) numbers: numbers that are divisible by the sum of their digits.

The main body of the sound has medium-pitch and sounds like light static. There is a prominent, ringing noise that gets subtly quieter as the sound progresses. Not audibly choppy.

Residue counts:
2   [72930, 22497]
3   [66716, 14519, 14192]
4   [47574, 11429, 25356, 11068]
5   [33440, 15354, 17151, 14150, 15332]
6   [50773, 3430, 11068, 15943, 11089, 3124]
7   [21770, 12288, 12263, 12264, 12305, 12293, 12244]
8   [28455, 5737, 12722, 5541, 19119, 5692, 12634, 5527]
9   [40474, 5197, 5310, 11880, 4764, 4575, 14362, 4558, 4307]
10   [26749, 4476, 13416, 4192, 11929, 6691, 10878, 3735, 9958, 3403]
11   [13791, 8254, 8012, 8348, 7951, 8363, 7937, 8356, 8002, 8321, 8092]
12   [32637, 1812, 3246, 7871, 7115, 1545, 18136, 1618, 7822, 8072, 3974, 1579]

Density information:
A(x) at x=10,100,1000,...:
[10, 33, 213, 1538, 11872, 95427]

A(x)/x at x=10,100,1000,...:
1.0,0.33,0.213,0.1538,0.11872,0.095427,

First 100 first differences:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 2, 1, 3, 3, 3, 6, 4, 2, 3, 3, 2, 4, 6, 3, 7, 2, 8, 1, 3, 6, 10, 2, 6, 2, 1, 1, 2, 3, 3, 6, 6, 1, 2, 5, 4, 6, 2, 1, 3, 6, 9, 9, 10, 2, 3, 3, 2, 1, 3, 3, 2, 1, 6, 4, 2, 2, 1, 3, 2, 4, 6, 3, 4, 5, 9, 3, 2, 4, 10, 5, 3, 12, 6, 2, 4, 3, 5, 2, 2, 6, 3, 3, 6, 9, 9, 4, 6, 2, 3]

First 99 second differences:
[0,0,0,0,0,0,0,0,1,4,-4,-1,2,0,0,3,-2,-2,1,0,-1,2,2,-3,4,-5,6,-7,2,3,4,-8,4,-4,-1,0,1,1,0,3,0,-5,1,3,-1,2,-4,-1,2,3,3,0,1,-8,1,0,-1,-1,2,0,-1,-1,5,-2,-2,0,-1,2,-1,2,2,-3,1,1,4,-6,-1,2,6,-5,-2,9,-6,-4,2,-1,2,-3,0,4,-3,0,3,3,0,-5,2,-4,1,]

There are dark horizontal spectral lines at 4411Hz, 4895Hz, 9797Hz, and 14700Hz. Dividing the sample rate 44100 by these frequencies gives values that are near to 10, 9, 4.5, and exactly 3 respectively. Noticeably, 4.5 and 3 are common fractions of 9, so these additional spectral lines may provide further evidence of a connection to modulo 9. The clearest residue-count connections are for mod 3 and mod 9. For mod 3, residue 0 has 66716 terms, while residues 1 and 2 have only 14519 and 14192 terms. For mod 9, residue 0 also has a much larger quantity of terms than the other residue classes. This is meaningful because dividing a number by 3 or 9 leaves the same remainder as dividing its digit sum by 3 or 9. For example, 738 has digit sum 7 + 3 + 8 = 18. Since 738 is divisible by 3 and 9, its digit sum 18 is also divisible by 3 and 9, and vice versa. This arithmetic fact helps explain the large number of terms congruent to 0 modulo 3 and modulo 9. In relation to integer 10, one observation is that modulo 10 is highly non-uniform as well. Residue 0 contains 26749 terms, while some residue classes contain fewer than 4000 terms. 

To prove that the sequence is infinite, we can use the example of powers of 10. The numbers 10, 100, 1000, and so on all have digit sum 1. Thus, every power of 10 is divisible by its digit sum. This subsequence proves that the sequence is infinite because there are infinitely many powers of 10. The values of A(x)/x decrease as x increases, which suggests that Niven numbers become less dense among larger integers.

-- Lauren Yee