Aliquot sum s(n) = sum of proper divisors of n. s(12)=1+2+3+4+6=16. Iterating s gives aliquot sequences: 12→16→15→9→4→3→1→0. Perfect: s(n)=n (cycle length 1). Amicable: s(a)=b, s(b)=a (cycle length 2). Open: does every sequence terminate?
How to Use
Enter n:
- s(n): Aliquot sum
- Sequence: Iterated sums
- Fate: Terminates, cycles, or grows
Catalan-Dickson Conjecture
Every aliquot sequence either terminates at 0, or enters a cycle (perfect, amicable, or sociable). The Lehmer five (276, 552, 564, 660, 966) have been iterated millions of terms without resolution!
Sequence Fates
- Terminate at 0: most numbers
- Fixed point: perfect (6,28,496)
- 2-cycle: amicable (220↔284)
- k-cycle: sociable (rare)
- Unknown: possibly unbounded
Step-by-Step Instructions
- 1Enter number.
- 2Compute s(n).
- 3Generate sequence.
- 4Detect cycles.
- 5Classify behavior.