Superabundant number: n where σ(n)/n > σ(k)/k for all k < n. Record holders for the abundancy index. First: 1,2,4,6,12,24,36,48,60,120,180,240,360... Robin: σ(n) < e^γ·n·ln(ln(n)) for n>5040 iff Riemann hypothesis is true.
How to Use
Enter n:
- SA?: Is σ(n)/n a new record?
- Ratio: σ(n)/n value
- Robin: σ(n) vs e^γ·n·ln(ln(n))
Robin's Inequality & RH
Robin (1984): the Riemann hypothesis is equivalent to σ(n) < e^γ·n·ln(ln(n)) for all n > 5040, where γ = 0.5772... (Euler-Mascheroni). Any counterexample must be a colossally abundant number.
Structure
Superabundant numbers have the form 2^a₁·3^a₂·5^a₃·...·pₖ^aₖ where a₁ ≥ a₂ ≥ ... ≥ aₖ ≥ 1. They use consecutive primes with non-increasing exponents. This maximizes σ(n)/n efficiently.
Step-by-Step Instructions
- 1Enter number.
- 2Check superabundant.
- 3View σ(n)/n.
- 4Test Robin's inequality.
- 5Browse sequence.