WXML Winter 2018

Number Theory and Noise

Nile Wymar, Sherry Chen, Robert Pedersen, Hongyu Cao

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 iddescription sound player creator
n such that the sum of digits of n^3 is one more than a cube downloadNile Wymar
n such that sum of digits of n^4 is a fourth power downloadNile Wymar
n such that sum of digits of n^2 equals 54 downloadNile Wymar
n such that sum of digits of n^3 equals 80 downloadNile Wymar
n such that sum of digits of n^4 equals 99 downloadNile Wymar
A004207 a(0) = 1, a(n) = sum of digits of all previous terms downloadNile Wymar
A004207 A004207 beginning at 6666706 downloadNile Wymar
A004207 A004207 beginning at 10000000042 downloadNile Wymar

A000057 Primes dividing all Fibonacci sequences. downloadSherry Chen
A000059 Numbers n such that (2n)^4 + 1 is prime. downloadSherry Chen
A000062 A Beatty sequence: a(n) = floor(n/(e-2)). downloadSherry Chen
A000068 Numbers n such that n^4 + 1 is prime. downloadSherry Chen
A000069 Odious numbers: numbers with an odd number of 1's in their binary expansion. downloadSherry Chen
A000099 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record. downloadSherry Chen
A000123 Number of binary partitions: number of partitions of 2n into powers of 2. downloadSherry Chen
A000124 Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts. downloadSherry Chen
A000125 Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1. downloadSherry Chen
A034048 Numbers with multiplicative digital root = 0. downloadSherry Chen
A277061 Numbers with multiplicative digital root > 0. downloadSherry Chen
A227510 Numbers such that product of digits of n is positive and a substring of n. downloadSherry Chen
A236402 Numbers with property that the sum of any pair of adjacent digits is a substring of the number. downloadSherry Chen
A254621 Zerofree numbers having product of digits less than or equal to sum of digits downloadSherry Chen
A062996 Sum of digits is greater than or equal to product of digits. downloadSherry Chen

A000977Numbers that are divisible by at least three different primes. downloadRobert Pedersen
A002971Numbers k such that 4*k^2 + 25 is prime. downloadRobert Pedersen
A003151Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))). downloadRobert Pedersen
A003152A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))). downloadRobert Pedersen
A003153 a(n) = integer nearest n*(1+sqrt(2)). downloadRobert Pedersen
A005100 Deficient numbers: numbers n such that sigma(n) < 2n. downloadRobert Pedersen
A005101 Abundant numbers (sum of divisors of n exceeds 2n). downloadRobert Pedersen
A005574 Numbers k such that k^2 + 1 is prime. downloadRobert Pedersen
A022843 Beatty sequence for e: a(n) = floor(n*e). downloadRobert Pedersen
A080081 Beatty sequence for (3+sqrt(13))/2. downloadRobert Pedersen
A022839 Beatty sequence for sqrt(5). downloadRobert Pedersen
A007304 Sphenic numbers: products of 3 distinct primes. downloadRobert Pedersen
A007064 Numbers not of form "nearest integer to n*tau", tau=(1+sqrt(5))/2. downloadRobert Pedersen
A110117 Beatty sequence for sqrt(2) + sqrt(3). downloadRobert Pedersen
3 seconds each of Beatty sequences for sqrt(2),sqrt(3),sqrt(5),sqrt(6), sqrt(7) downloadRobert Pedersen
1 seconds each of Beatty-like sequences based on c.f. convergents of sqrt(5) downloadRobert Pedersen

back to main project page