On the smallest abundant number not divisible by the first $k$ primes

Abstract

We say a positive integer $n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the positive divisors of $n$. Number the primes in ascending order: $p_1=2$, $p_2=3$, and so forth. Let $A(k)$ denote the smallest abundant number not divisible by $p_1$, $p_2$, \dots, $p_k$. In this paper we find $A(k)$ for $1\leq k\leq 7$, and we show that for all $\epsilon>0$, $(1-\epsilon)(k\ln{k})^{2-\epsilon}<\ln{A(k)}<(1+\epsilon)(k\ln{k})^{2 +\epsilon}$ for all sufficiently large $k$.