*Headphones are strongly recommended for listening to the sounds on this page.*

For a given integer sequence (by which I mean an increasing sequence of positive integers), we can make an audio file by creating audio events at times corresponding to the terms of the sequence.

If the sequence is {a_{1}, a_{2}, ...}, then we can create sound events at times
{*r* a_{1}, *r* a _{2}, ...} for some suitable scaling factor, *r*.

Depending on the value of *r* and the nature of the sound events created,
many results are possible.

Here, I have taken this to an extreme. The sound events are one sample in length,
of maximal amplitude, and *r* is 1/44,100 of a second, corresponding to the
standard CD sound sampling rate.

Thus, the audio file created has samples that are merely "on" or "off". The n-th sample
is on only if *n* appears in the sequence.

The result is "noise", in most cases, with characteristics which depend on the nature of the sequence.

A curious feature of this method is that the resulting sounds from a sequence, and from the complement of that sequence, sound identical, since one is the inverted waveform of the other. Harmonic components can appear due to the presence, or non-presence, of certain multiples.

When a sequence is not too "regular", and doesn't thin out too quickly (as, e.g., the squares sequence below illustrates), we may get something interesting.

- Primes less than 10
^{6}(23 sec.) This illustrates the randomness of the primes, and yet it also shows their consistency: it's noise, but it's essentially always the same noise. - Primes from 10
^{20}to 10^{20}+10^{6}(23 sec.) The primes do thin out, but it takes a long time to notice it. You can't actually hear the thinning at the rate these files are made, so we jump ahead to 10^{20}, and we hear the noise is now more of a roar: a bit quieter, and more crackly, as the primes are less dense among the integers here. - Semiprime numbers (i.e. numbers which are the product of two distinct primes) less than 10
^{6}(23 sec.) Semiprimes are more common than primes, so this is louder and smoother than the prime sounds. Strong harmonics include those at 44100/4=11025 and 44100/5 = 8820, since no multiples of 4 are in the sequence, and all prime multiples of 5 are. - Abundant numbers less than 10
^{6}(23 sec.) Every multiple of 6 is abundant, so we hear harmonic peaks at 44100/6=7350, 7350/2=3675, 7350/3=2450, etc. This makes the sound less pure noise in a certain sense, and more of a buzz or machine type sound. - Ten minutes of the abundant numbers (less than 27 million) (612 sec.)
- Square-free numbers less than 10
^{6}(23 sec.) Square-free numbers have a positive asymptotic density of 6/pi^{2}, so they don't thin out. The sound is thick and quite stable. Since multiples of 4, 9, 25, 36, etc., are not among this sequence, there are harmonic peaks at 44100/4,44100/9,44100/25, etc., and multiples of these frequencies. - Numbers whose number of divisors is a power of 2, less than 10
^{6}(23 sec.) If a number is square-free, then it has a number of divisors that is a power of 2. However, other numbers also have this property (2^{7}, for instance). This sound doesn't have quite as strong peaks as the square-free one; the most prominent below 11,000 is 5512=44100/8. Note that 8 times a square-free odd number is a number in this sequence. - Numbers with exactly 6 divisors, less than 10
^{6}(23 sec.) Quite similar to the primes sound. - Number with exactly 10 divisors, less than 10
^{6}(23 sec.) Also quite similar to the primes sound. - Numbers which are equal to σ(n) for some n, less than 10
^{6}(23 sec.) The function σ(n) is the sum of divisors of n. For example, &sigma(4)=1+2+4=7. So 7 is part of this sequence. The number 5, however, is not: there is no number n such that σ(n)=5. This sequence is pretty uniformly noisy, but there is a peak at 7350 hz, since 44100/7350=6 and lots of values of σ(n) are multiples of 6. - Ulam numbers (1, 2, 3, 4, 6, 8, 11, ...) less than 10
^{6}(23 sec.) This one has a strong harmonic peak at about 2040 hz (44100/2040 = 21.617...). The Ulam numbers have a known wavelike nature, with repeated clumps (see, for instance, the comments here). - Happy numbers less than 10
^{6}(23 sec.) This one has a lot going on in it. Definitely one for headphones. - Harshad numbers less than 10
^{6}(23 sec.) You can clearly hear the effect of this being a base-dependent sequence, with a distinct cycling every 100,000, and a weaker one every 10,000. A strong harmonic component at 4900=44100/9 hz. - Palindromic numbers, less than 10
^{6}(23 sec.) Palindromes are rather rare, so the sound is mostly "clicking" rather than noise. - Squares less than 10^6 (23 sec.) Illustrates what happens if the sequence has a very regular distribution. Not noisy at all.