Deficient: σ(n)−n < n. All primes p are deficient (σ(p)=p+1, deficiency p−1). All prime powers p^k are deficient. About 75.2% of natural numbers are deficient. The deficiency of n is 2n−σ(n).
How to Use
Enter a number:
- Classification: Deficient/Perfect/Abundant
- σ(n): Divisor sum
- Deficiency: 2n−σ(n)
Primes & Powers
Every prime p has σ(p)=1+p, so deficiency = p−1. Every prime power p^k has σ = (p^{k+1}−1)/(p−1) < 2p^k. So ALL prime powers are deficient. Powers of 2 have σ(2^k) = 2^{k+1}−1, deficiency just 1!
Density
~75.2% of natural numbers are deficient. Combined with ~0.0...% perfect and ~24.8% abundant, this accounts for all positive integers. Deficient numbers are the 'majority' — abundant numbers are special.
Step-by-Step Instructions
- 1Enter number.
- 2Compute σ(n).
- 3Check deficiency.
- 4See divisors.
- 5Classify.