HCN Exponent Explorer · affine·mess

affine·mess

exponent gain loss

n = 1 τ = 1 digits 1 ω = 0 frame 5 × 4

a(1) = 1

Highly composite numbers

A positive integer is highly composite if it has more divisors than every smaller positive integer: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, … Ramanujan wrote the founding paper on them in 1915.

What you're looking at

Each bar is the exponent of one prime in a(n), the n-th highly composite number. Scrub or press play to walk the sequence. A pale cap marks an exponent that grew since the previous term; an orange outline marks one that shrank.

The exponents never increase as the primes increase: if a prime appeared with a larger exponent than a smaller prime, swapping them would give a smaller number with the same divisor count. That's why the picture is always a staircase. Ramanujan also showed the last exponent is 1, except for 4 and 36.

The frame widens and heightens in fixed steps and never shrinks back, so you can watch the staircase stretch out as new primes arrive.

Keys

← / → step one term (Shift: ten). Space plays and pauses. Hover a bar for its prime and exponent.

Data

The first 10,000 terms of OEIS A002182, factored exactly. Prime index runs 1…179 (2 … 1063); exponents run 0…16. τ is the number of divisors, ω the number of distinct primes.