4,104

4 + 1 + 0 + 4

Digit sum repeated until one digit remains; equals n mod 9 (with 9 for multiples of 9).

Reverses to 4,014.

Divisible by 2.

Composite — it factors as 2^3 × 3^3 × 19.

3 distinct prime factors, 7 with multiplicity.

At least one prime appears squared.

Zero, because a squared prime divides n.

A gap of 12 between them.

1, 2, 3, 4, 6, 8, 9, 12, 18, 19, 24, 27, 36, 38, 54, 57, 72, 76, 108, 114, … (12 more)

The proper divisors — everything except n itself.

Its proper divisors overshoot by 3,792.

How many integers from 1 to n share no factor with n.

A prime congruent to 3 mod 4 divides it an odd number of times, which rules it out.

Legendre's three-square theorem allows it.

Lagrange's four-square theorem: every non-negative integer is, without exception.

Expressible in 2 distinct ways — a taxicab number, after the Hardy–Ramanujan story about 1729.

The smallest pair of primes summing to n. Goldbach’s conjecture says one always exists; it remains unproven.

Not triangular, square, pentagonal, hexagonal, cubic, Fibonacci, Lucas, Catalan, factorial or a power of two.

Summing the squares of its digits falls into the 4 → 16 → 37 → … → 4 cycle.

Divisible by its digit sum 9 — 4,104 ÷ 9 = 456.

Peaks at 4,104 along the way.

Standard Roman numerals only run from 1 to 3,999.