172

1 + 7 + 2

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

Reverses to 271.

Divisible by 2.

Composite — it factors as 2^2 × 43.

2 distinct prime factors, 3 with multiplicity.

At least one prime appears squared.

Zero, because a squared prime divides n.

A gap of 6 between them.

1, 2, 4, 43, 86, 172

The proper divisors — everything except n itself.

Its proper divisors fall short by 36.

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.

Not divisible by its digit sum 10.

Peaks at 196 along the way.