6

6

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

Reads the same in both directions.

Divisible by 2.

Composite — it factors as 2 × 3.

2 distinct prime factors, 2 with multiplicity.

No prime divides it twice.

A product of exactly two primes, counted with multiplicity.

(−1) to the power of 2 distinct prime factors.

A gap of 2 between them.

1, 2, 3, 6

The proper divisors — everything except n itself.

Its proper divisors sum to exactly itself.

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.

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

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

Every single-digit number equals itself raised to the first power, so all ten qualify trivially.

Divisible by its digit sum 6 — 6 ÷ 6 = 1.

Its square 36 ends in 6.

Peaks at 16 along the way.