119

1 + 1 + 9

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

Reverses to 911.

Leaves remainder 1 on division by 2.

Composite — it factors as 7 × 17.

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 14 between them.

1, 7, 17, 119

The proper divisors — everything except n itself.

Its proper divisors fall short by 94.

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 11.

Peaks at 808 along the way.