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The classic sequence: 2, 3, 5, 7, 11, 13… Each prime triggers a sine tone. Pitch wraps around the pentatonic scale via (prime mod scale_size). Primes grow sparser—the silences between tones widen as N increases.
The complement of the primes: 4, 6, 8, 9, 10, 12… Uses a square wave oscillator—sharper, more mechanical than the primes' sine. The density is high and the sound feels crowded, which is mathematically appropriate.
Prime pairs (p, p+2) where both are prime: (3,5), (5,7), (11,13), (17,19)… Each pair plays as a two-note chord with a brief 60ms delay between voices—a pluck-pluck. It is an open conjecture that infinitely many twin primes exist.
The gap g_n = p_{n+1} - p_n between consecutive primes. For most small n the gap is 2 (twin primes); occasional large gaps (like g(23)=6, or Bertrand's famous large gaps) sound as high, isolated notes.
Goldbach's conjecture (1742, unproven): every even integer > 2 equals the sum of two primes. For each even n, find a representation n = p + q and play p then q in quick succession. The two tones converge as p and q balance toward n/2.
The ancient algorithm: start from 2, mark all its multiples as composite, advance to the next unmarked number, repeat. Crossing-out composites plays a dry click; surviving primes ring a sine tone. Watch the canvas—the crossed numbers appear as the sieve sweeps through.
Primes of the form 2^p - 1: 3, 7, 31, 127, 8191, 131071, 524287… They are extraordinarily rare—only 51 are known as of 2024. This mode plays the known small Mersenne primes in sequence with a deep, sustained bass tone. Each one feels like a slow bell.
From any starting seed, repeatedly apply: if even, divide by 2; if odd, multiply by 3 and add 1. The trajectory always (conjectured) reaches 1. The pitch traces the trajectory value; the trajectory from seed 27 takes 111 steps and peaks at 9232 before crashing to 1.