WXML Winter 2020

Number Theory and Noise

Thien Do, Pubo Huang, Ally Krinsky,

This project investigates the representations of sets of positive integers (sequences) as sound.

A digital audio waveform is created from a given set A of positive integers by setting sample number i to a non-zero constant c for all i in the set. All other samples are set to zero.

For example, the waveform for the primes starts like this:



We use the standard CD-audio sampling rate of 44100 samples per second, so Δt = 1/44100= 0.0000226757... seconds.

For many sets, the result is what most people would describe as noise.

OEIS sequences

OIES id description sound player download creator quarter tags comments
A000037 Numbers that are not squares (or, the nonsquares). download Ally Krinsky Win2020
A002328 Numbers n such that n^2-n-1 is prime. download Ally Krinsky Win2020
A002642 Numbers n such that (n^2+n+1)/3 is prime. download Ally Krinsky Win2020
A002643 Numbers n such that (n^2+n+1)/19 is prime. download Ally Krinsky Win2020
A002644 Numbers n such that (n^2+n+1)/21 is prime. download Ally Krinsky Win2020
A002731 Numbers n such that (n^2+1)/2 is prime. download Ally Krinsky Win2020
A002733 Numbers n such that (n^2 + 1)/10 is prime. download Ally Krinsky Win2020
A002970 Numbers n such that 4*n^2 + 9 is prime. download Ally Krinsky Win2020
A004083 Numbers n such that cos(n-1) <= 0 and cos(n) > 0. download Ally Krinsky Win2020
A002640 Numbers n such that (n^2 + n + 1)/3 is prime. download Ally Krinsky Win2020
A002641 Numbers n such that (n^2 + n + 1)/7 is prime. download Ally Krinsky Win2020
A002642 Numbers n such that (n^2 + n + 1)/7 is prime. download Ally Krinsky Win2020
A002643 Numbers n such that (n^2 + n + 1)/7 is prime. download Ally Krinsky Win2020
A004614 Numbers that are divisible only by primes congruent to 3 mod 4. download Ally Krinsky Win2020
A007519 Primes congruent to 1 mod 8. download Ally Krinsky Win2020 primes
A007520 Primes congruent to 3 mod 8. download Ally Krinsky Win2020 primes
A007521 Primes congruent to 5 mod 8. download Ally Krinsky Win2020 primes
A001000 a(n) = least m such that if a/b < c/d where a,b,c,d are integers in [0,n], then a/b < k/m < c/d for some integer k. download Thien Do Win2020
A001748 3*primes. download Thien Do Win2020
A001750 5*primes. download Thien Do Win2020
A001915 Primes p such that the congruence 2^x = 3 (mod p) is solvable. download Thien Do Win2020 primes
A002970 Numbers n such that 4*n^2 + 9 is prime. download Thien Do Win2020
A100317 Numbers k such that exactly one of k - 1 and k + 1 is prime. download Thien Do Win2020
A004759 Binary expansion starts 111. download Thien Do Win2020
A006532 Numbers n such that sum of divisors of n is a square. download Thien Do Win2020
A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2. download Thien Do Win2020
A015782 Phi(n) + 3 divides sigma(n + 3). download Thien Do Win2020
A024917 a(n) = Sum_{k=2..n} k*floor(n/k). download Thien Do Win2020

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