A weird number is abundant (its proper divisors sum to more than the number) yet not semiperfect (no subset of those divisors hits the number exactly). The divisors overshoot but can never land.
The smallest weird number is 70: its proper divisors 1, 2, 5, 7, 10, 14, 35 sum to 74 > 70, but no subset sums to exactly 70. The next are 836, 4030, 5830, 7192.
Weird numbers are rare and mysterious: all known weird numbers are even, and whether an odd weird number exists is an open problem (none below 10²¹). Erdős offered a cash prize for settling it. Stanley Benkoski coined the name in 1972, and a Benkoski–Erdős paper proved weird numbers have positive density — rare, but never running out.