Abundant: σ(n)−n > n, i.e., proper divisors sum to more than n. 12 is the smallest (1+2+3+4+6=16>12). All multiples of 6 beyond 6 are abundant. All multiples of abundant numbers are abundant. The density of abundant numbers approaches ~24.76%.
How to Use
Enter a number:
- Classification: Abundant/Perfect/Deficient
- σ(n): Divisor sum
- Abundance: σ(n)−2n
Goldbach-like
Every integer > 20161 is a sum of two abundant numbers. This was proven by Schur (1930s). The largest number NOT expressible as a sum of two abundant numbers is 20161.
Density
The natural density of abundant numbers is between 0.2474 and 0.2480 (about 24.8%). This means roughly 1 in 4 natural numbers is abundant. The odd abundant numbers are rarer — the smallest is 945.
Step-by-Step Instructions
- 1Enter number.
- 2Compute σ(n).
- 3Check abundance.
- 4See divisors.
- 5Classify.