175

1 + 7 + 5

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

Reverses to 571.

Leaves remainder 1 on division by 2.

Composite — it factors as 5^2 × 7.

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, 5, 7, 25, 35, 175

The proper divisors — everything except n itself.

Its proper divisors fall short by 102.

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.

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 13.

Peaks at 9,232 along the way.