## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Douglas E. Iannucci

#### 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 $k$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin Volume 12, Number 1 (2005), 39-44.

Dates
First available in Project Euclid: 12 April 2005

http://projecteuclid.org/euclid.bbms/1113318127

Mathematical Reviews number (MathSciNet)
MR2134854

Zentralblatt MATH identifier
02186501

Subjects
Primary: 11A32 11Y70: Values of arithmetic functions; tables

Keywords
abundant numbers primes

#### Citation

Iannucci, Douglas E. On the smallest abundant number not divisible by the first $k$ primes. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 1, 39--44. http://projecteuclid.org/euclid.bbms/1113318127.