# Number Theory and Noise

## Jingyun Du, Pubo Huang, Jin Lin, Sam Wang

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
A008846 Hypotenuses of primitive Pythagorean triangles download Jin Lin Aut2019
A000788 Total number of 1's in binary expansions of 0, ..., n. download Jin Lin Aut2019 digits
A054632 Partial sums of the sequence of digits of the natural numbers: 1,2,3,..,9,1,0,1,1,1,2,1,3,1,4,... download Jin Lin Aut2019 digits
A067112 Partial sums of the sequence of digits of the primes (2,3,5,7,1,1,1,3,...) download Jin Lin Aut2019 digits
A067113 Partial sums of the sequence of digits of the squares (1,4,9,1,6,2,5,...) download Jin Lin Aut2019 digits
Partial sums of the sequence of digits of n^3 download Jin Lin Aut2019 digits
Partial sums of the sequence of digits of n^4 download Jin Lin Aut2019 digits
Partial sums of the sequence of digits of n^5 download Jin Lin Aut2019 digits
Partial sums of the sequence of digits of n^n download Jin Lin Aut2019 digits
Partial sums of digits in decimal expansion of 2^sqrt(2) download Jin Lin Aut2019 digits
A099539 Partial sums of the digits in decimal expansion of sqrt(2) download Jin Lin Aut2019 digits
A099534 Partial sums of the digits in decimal expansion of e download Jin Lin Aut2019 digits
Partial sums of the digits in decimal expansion of sqrt(3) download Jin Lin Aut2019 digits
A067113 Partial sums of the sequence of digits of the squares (1,4,9,1,6,2,5,...) download Jin Lin Aut2019 digits
A225771 Numbers that are positive integer divisors of 1 + 2*x^2 where x is a positive integer download Jin Lin Aut2019
A000408 Numbers that are the sum of three nonzero squares. download Jin Lin Aut2019
A000404 Numbers that are the sum of 2 nonzero squares. download Jin Lin Aut2019
A000379 Numbers n where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: download Jin Lin Aut2019
A000328 Number of points of norm <= n^2 in square lattice. download Jin Lin Aut2019
A164620 Primes p such that 1 +p*floor(p/2) is also prime. download Pubo Huang Aut2019 primes
A164624 Primes p such that p+floor(p/2)+floor(p/3) is prime. download Pubo Huang Aut2019 primes comment
A164625 Primes p such that p+floor(p/2)+floor(p/3)+floor(p/5) is prime. download Pubo Huang Aut2019 primes
A000960 Flavius Josephus's sieve numbers. download Pubo Huang Aut2019
A046959 Numbers n where sin(n) increases monotonically to 1. download Pubo Huang Aut2019
A093083 Partial sums of digits of decimal expansion of golden ratio, phi. download Pubo Huang Aut2019 digits
Partial sums of digits of randomly-generated decimal expansion. download Pubo Huang Aut2019 digits
Numbers n for which the digit sum of n^2 is 28 download Pubo Huang Aut2019 digits
Numbers n for which the digit sum of n^2 is 31 download Pubo Huang Aut2019 digits
Numbers n for which the digit sum of n^2 is 72 download Pubo Huang Aut2019 digits
A259368 Number of binary digits of n^n. download Pubo Huang Aut2019 digits
floor( n atan(n/10^3)) download Pubo Huang Aut2019
floor( n atan(n/10^4)) download Pubo Huang Aut2019
floor( n atan(n/10^5)) download Pubo Huang Aut2019
strictly increasing portion of floor( n atan(n/10^6)) download Aut2019
A000430 Primes and squares of primes. download Sam Wang Aut2019
A000452 The greedy sequence of integers which avoids 3-term geometric progressions. download Sam Wang Aut2019
A000469 1 together with products of 2 or more distinct primes. download Sam Wang Aut2019
A000549 Numbers that are the sum of 2 squares but not sum of 3 nonzero squares. download Sam Wang Aut2019
A000966 n! never ends in this many 0's. download Sam Wang Aut2019
A001043 Numbers that are the sum of 2 successive primes. download Sam Wang Aut2019
A001101 Moran numbers: n such that (n / sum of digits of n) is prime. download Sam Wang Aut2019
A001122 Primes with primitive root 2. download Sam Wang Aut2019
A001123 Primes with 3 as smallest primitive root. download Sam Wang Aut2019
A007624 Product of proper divisors of n = n^k, k>1. download Sam Wang Aut2019
A007634 Numbers n such that n^2 + n + 41 is composite. download Sam Wang Aut2019
A007638 Numbers k such that 3*k^2 - 3*k + 23 is composite. download Sam Wang Aut2019
A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways. download Sam Wang Aut2019
A007774 Numbers that are divisible by exactly 2 different primes. download Sam Wang Aut2019
A003277 Cyclic numbers: n such that n and phi(n) are relatively prime; download Sam Wang Aut2019
A003159 Numbers n whose binary representation ends in an even number of zeros. download Sam Wang Aut2019
A002035 Numbers that contain primes to odd powers only. download Sam Wang Aut2019 primes
A001958 floor( (n+1/3) * (5+sqrt(13)) /2) download Sam Wang Aut2019
A001958 approximation: difference sequence is 5, 4, 4, 4, 5, 4, 4, 5, 4, 4, then repeats. download Sam Wang Aut2019
A001969 Evil numbers: numbers with an even number of 1's in their binary expansion. download Jingyun Du Aut2019
A027699 Evil primes: primes with even number of 1's in their binary expansion. download Jingyun Du Aut2019
A125494 Composite evil numbers. download Jingyun Du Aut2019
A014574 Average of twin prime pairs. download Jingyun Du Aut2019
A045718 Nearest neighbors of primes. download Jingyun Du Aut2019
A088485 Numbers n such that n^2 + n - 1 and n^2 + n + 1 are twin primes. download Jingyun Du Aut2019
A111980 Union of pairs of consecutive primes p, q with q-p = 4. download Jingyun Du Aut2019