Practical number: every integer 1 to n is a sum of distinct divisors of n. Example: 12 with divisors {1,2,3,4,6,12} — can make every number 1-12. First: 1,2,4,6,8,12,16,18,20,24,28,30,32... All powers of 2 are practical. All even perfect numbers are practical.
How to Use
Enter n:
- Practical?: Can represent all 1..n
- Criterion: Srinivasan test
- Examples: Show representations
Srinivasan's Criterion
n = p₁^a₁ · p₂^a₂ ·...· pₖ^aₖ (primes in order). n is practical iff for each i: pᵢ ≤ 1 + σ(p₁^a₁ · ... · pᵢ₋₁^aᵢ₋₁). This gives an O(log n) test!
Properties
- Density ~c·n/√(ln n) for constant c
- Even practical numbers have same density as primes asymptotically
- Goldbach-like: every even ≥ 2 = sum of two practical numbers
Step-by-Step Instructions
- 1Enter number.
- 2Check practical.
- 3See criterion.
- 4View representations.
- 5Browse sequence.