Audible Number Sequences
Just for the fun of it (and to get some useful raw material for weird noise mucsical experiments) I took a bunch of number sequences, computed the first million entries, slapped them normailzed into a sound file @ 16kHz and see what happens. Some of them are quite nice. Enjoy.
1,000,000 samples at 16 kHz results in 1:02.5 minutes of sound and for each sound there also is a picture of the waveform. If the waveform of the whole sound would reduce to a solid block of samples I increased the spatial resolution (i.e. zoomed in) to give more information.
Caveat: Some of the sounds are quite harsh and their levels are very different, so mind your speakers/ears.
Euler's Pentagonal Theorem Sample(n) = (-1)^m if n is of the form m(3m+-1)/2, otherwise a(n)=0 Sloane's A010815
Goldbach decompositions Sample(n) = the number of decompositions of 2n into an unordered sum of two odd primes The number of Goldbach decompositions is strictly increasing on the long run (well, most likely), so it starts pretty soft. Sloane's A002375
Moebius function Sample(1) = 1; Sample(n) = (-1)^k if n is the product of k different primes; otherwise Sample(n) = 0 Sloane's A008683
Merten's function Sample(n) = Sum{1<=k<=n} mu(k), where mu is the Moebius function The partial sum of the moebius function produces a wind like sound, albeit pretty quite. Sloane's A002321
Nearest integer to sin() Sample(n) = round(sin(n)) Creates several overlayed frequencies. Sloane's A000494
Nørgård's Infinity Sequence Sample(0)=0, Sample(2n) = -Sample(n), Sample(2n+1) = Sample(n) + 1 Invented in an attempt to unify in a perfect way repetition and variation. This was probably not what he invented it for... Sloane's A004718
Primerace modulo 3 Sample(n) = (Number of 3k-1 primes <= n) - (Number of 3k+1 primes <= n) Also known as Chebyshev Bias. Sloane's A038698
Interestingly this produces a pretty perfect pink noise...
Primerace modulo 4 Sample(n) = (Number of 4k-1 primes <= n) - (Number of 4k+1 primes <= n) This produces an almost identical pink noise to primerace mod 3. Sloane's A038698
Surfeit of oddly factored numbers over evenly factored numbers Sample(n) = (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n) This also produces a cool pink noise. Sloane's A072203
Smallest distance to prime power Sample(n) = min(abs(n-k)) : where k runs through the prime powers Sloane's A080732
Somehow the OEIS doesn't seem to list the non-absolute version which also sounds quite nice: Sample(n) = min_of_abs(n-k) : where k runs through the prime powers
Zero bits minus one bits Sample(n) = (Number of 0's)-(number of 1's) in base 2 representation of n Sloane's A037861
Legendre symbol (n,31) Sample(n) = legendre(n, 31) See description below
Partial Sum of Legendre symbol (n, 31) Sample(n) = Sum{1<=k<=n} legendre(k, 31) To be honest, this sound is not particularly interesting, I just wanted to have a Legendre sequence in it with a lower modulus for comparison. Oh, and I liked the shape of this waveform. Now if I could find another one that is penguin shaped or so...
Legendre symbol (n,257) Sample(n) = legendre(n, 257) Sloane's A165573
Partial Sum of Legendre symbol (n, 257) Sample(n) = Sum{1<=k<=n} legendre(k, 257) Sloane's A165575
Legendre symbol (n,263) Sample(n) = legendre(n, 263) Sloane's A165574
Partial Sum of Legendre symbol (n, 263) Sample(n) = Sum{1<=k<=n} legendre(k, 263) Sloane's A165576
Legendre symbol (n,1597) Sample(n) = legendre(n, 1597)
Partial Sum of Legendre symbol (n, 1597) Sample(n) = Sum{1<=k<=n} legendre(k, 1597)
Legendre symbol (n,8191) Sample(n) = legendre(n, 8191)
Partial Sum of Legendre symbol (n, 8191) Sample(n) = Sum{1<=k<=n} legendre(k, 8191)
Legendre symbol (n,28657) Sample(n) = legendre(n, 28657)
Partial Sum of Legendre symbol (n, 28657) Sample(n) = Sum{1<=k<=n} legendre(k, 28657)
Legendre symbol (n,65537) Sample(n) = legendre(n, 65537) 65537 is the 4th Fermat prime. Sloane's A165471
Partial Sum of Legendre symbol (n, 65537) Sample(n) = Sum{1<=k<=n} legendre(k, 65537) 65537 is the 4th Fermat prime. Sloane's A165472