150

1 + 5 + 0

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

Reverses to 51.

Divisible by 2.

Composite — it factors as 2 × 3 × 5^2.

3 distinct prime factors, 4 with multiplicity.

At least one prime appears squared.

Zero, because a squared prime divides n.

A gap of 2 between them.

1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

The proper divisors — everything except n itself.

Its proper divisors overshoot by 72.

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.

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 6 — 150 ÷ 6 = 25.

Peaks at 340 along the way.