weird number

Given an abundant number $n$, if no subset of its divisors $d_1,\mathrm{\dots },d_{\tau (n)-1}$ (where $\tau$ is the divisor function) can be selected that adds up to $n$ (that is, $n$ is not a semiperfect number), then $n$ is said to be a weird number.

The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, (listed in A006037 of Sloane’s OEIS). All the known weird numbers are even. In 1972, Benkoski wondered if there are any odd weird numbers; to this day ${10}^{17}$ is an accepted lower bound. Despite this question of parity, it has been proven that there are infinitely many weird numbers and that they have positive Schnirelmann density.

Even so, weird numbers are rarer than semiperfect numbers; twice a weird number is usually a semiperfect number, which makes all subsequent multiples of a weird number also semiperfect.

0.1 Trivia

The band Boards of Canada records for the label Music70. Track 9 of their album Geogaddi is titled ”The Smallest Weird Number.”