Powerful number: if prime p divides n, then p² divides n. Equivalently: n = a²·b³ for some a,b ≥ 1. First: 1,4,8,9,16,25,27,32,36,49,64,72,81,100... All perfect powers are powerful. Density ~c·√n (Golomb). Connected to the ABC conjecture.
How to Use
Enter n:
- Powerful?: Is p|n ⟹ p²|n?
- Factorization: Full prime decomposition
- Form: n = a²·b³
ABC Conjecture
The ABC conjecture implies: only finitely many powerful numbers n with n+1 also powerful (consecutive powerful pairs). Known: (8,9), (288,289), (675,676)... The conjecture predicts these are extremely rare.
Density & Counting
Number of powerful integers ≤ x is ~(ζ(3/2)/ζ(3))·√x ≈ 2.173·√x. Much denser than perfect powers but sparser than squareful numbers. About 1 in √n numbers up to n are powerful.
Step-by-Step Instructions
- 1Enter number.
- 2Check powerful.
- 3View factorization.
- 4Find a²b³ form.
- 5Browse sequence.