Aliquot Sequence Calculator

s(n) = σ(n) − n iterated

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About Aliquot Sequence Calculator

An aliquot sequence calculator iterating s(n) = σ(n)−n (sum of proper divisors). Sequences can terminate at 0, reach a fixed point (perfect number), cycle (amicable/sociable), or diverge. Shows trajectory and classification. Client-side.

Aliquot Sequence Calculator Features

  • Sequence
  • Classification
  • Trajectory
  • Perfect/amicable
  • Chart
Aliquot sequence: iterate s(n) = sum of proper divisors. s(12)=1+2+3+4+6=16, s(16)=1+2+4+8=15, s(15)=1+3+5=9... Catalan's conjecture: all sequences eventually terminate or cycle. Open for many starting values (e.g., 276).

How to Use

Enter n:

  • Sequence: Iterated s(n)
  • Type: Terminates, cycles, or unknown
  • Length: Steps to termination

Sequence Types

  • Terminates at 0: most numbers
  • Fixed point: perfect numbers (6,28,496)
  • 2-cycle: amicable pairs (220↔284)
  • k-cycle: sociable numbers

Open Problems

The Catalan-Dickson conjecture says all sequences eventually terminate or cycle. The sequence starting at 276 has been computed for billions of terms without resolving. Five 'driver' sequences (276, 552, 564, 660, 966) remain open.

Step-by-Step Instructions

  1. 1Enter n.
  2. 2View sequence.
  3. 3Classify behavior.
  4. 4See trajectory.
  5. 5Check perfect/amicable.

Aliquot Sequence Calculator — Frequently Asked Questions

What happens to the sequence for most numbers?+

Most sequences terminate at 0 (reaching a prime, then 1, then 0). About 80% of starting values below 1000 terminate. The rest reach perfect numbers, amicable pairs, or remain unresolved.

Why is 276 famous?+

The aliquot sequence starting at 276 has been computed for over a billion terms, reaching numbers with hundreds of digits, without terminating or cycling. It's the smallest unresolved starting value. Guy and Selfridge conjectured it diverges to infinity.

What are sociable numbers?+

Numbers forming a cycle of length k>2 under iteration of s(n). The first known sociable chain has length 5: 12496→14288→15472→14536→14264→12496. Sociable chains of length 28 are also known.

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