On primitive weird numbers of the form $2^{k}pq$

Abstract

We say a natural number~$n$ is abundant if $\sigma (n)>2n$, where $\sigma (n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there is some subset of the aliquot parts of~$n$ which sum to~$n$. We say~$n$ is weird if~$n$ is abundant but not pseudoperfect. We call a weird number~$n$ primitive if none of its aliquot parts are weird. We find all primitive weird numbers of the form $2^{k}pq$ ($p<q$ being odd primes) for $1\le k\le 14$. We also find primitive weird numbers of the same form, larger than any previously published.