156

1 + 5 + 6

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

Reverses to 651.

Divisible by 2.

Composite — it factors as 2^2 × 3 × 13.

3 distinct prime factors, 4 with multiplicity.

At least one prime appears squared.

Zero, because a squared prime divides n.

A gap of 6 between them.

1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156

The proper divisors — everything except n itself.

Its proper divisors overshoot by 80.

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.

Excluded by Legendre: it has the form 4^a(8b + 7).

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 12 — 156 ÷ 12 = 13.

Peaks at 304 along the way.