Comments by Erik Huang:
A006753 Smith numbers:
composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity).
GP code:
isA006753(n) = if(isprime(n), 0, my(f=factor(n)); sum(i=1, #f[, 1], sumdigits(f[i,1])*f[i, 2]) == sumdigits(n))
for(i=1, 1e6, isA006753(i) && print (i))
Comments:
On the spectrogram, there's one significant spectral peak at 4894 Hz and
3 more peaks that are multiples of this frequency.
Dividing such frequency into our sample frequency and we get number 9.0
To find a relationship between number 9 and smith number sequence,
here is the counts in residue classes mod 9 up to 10^6:
[4368, 763, 1220, 2633, 10711, 903, 6497, 499, 2334]
Among these counts, (4 mod 9) is the most dominant one.
To explain this effect, we investigated in properties of prime factors and found:
For every smith number that has 2 prime factors, it is always congruent to 4 mod 9
For every smith number that has 3 prime factors, it is can be congruent to 3 mod 9 or 6 mod 9
....
Note that lots of small integers are made of 2 prime factors;
Having more prime factors also results in an increase in sum of digits.
This phenomena lasts even when numbers get large
We counted the residue classes mod 9 in the interval from 10^10 to 10^10+10^6:
[414, 128, 238, 381, 3413, 159, 759, 85, 400]