Highly composite numbers (HCN): n is HCN if d(n) > d(m) for all m < n. First few: 1,2,4,6,12,24,36,48,60,120,180,240,360,720,1260,2520... Also called 'anti-primes'. Ramanujan's 1915 paper studied them extensively.
How to Use
Enter limit or count:
- HCNs: List highly composites
- d(n): Divisor counts
- Factorization: Prime decomposition
Structure
HCNs have prime factorizations n = 2^a₁·3^a₂·5^a₃·... where a₁≥a₂≥a₃≥... (exponents non-increasing) and all primes up to the largest are used (no gaps). 360=2³·3²·5¹ follows this pattern.
Ramanujan
Ramanujan's 1915 paper gave a complete characterization. He also defined 'superior highly composite numbers' with even stronger properties. His work connected these to the Riemann hypothesis!
Step-by-Step Instructions
- 1Enter count.
- 2Generate HCNs.
- 3See divisor counts.
- 4Check factorizations.
- 5Compare.