Aliquot Process Introduction

• Aliquot process: take proper factors of number (all factors except number itself), add them up
• Example: 8 → factors 1,2,4 → sum = 7
• Etymology: "aliquot" from Latin meaning "part of something" (related to quote, quotient, alias)
• Aliquot sequence: repeatedly apply aliquot process to results

Number Classifications

• Deficient numbers: aliquot sum < original number (e.g., 8 → 7 → 1)
• Abundant numbers: aliquot sum > original number (e.g., 24 → 36)
  • 24 factors: 1,2,3,4,6,8,12 → sum = 36
• Perfect numbers: aliquot sum = original number
  • 6 → factors 1,2,3 → sum = 6 (stays in loop)
  • Other perfect numbers: 28, 496
  • No odd perfect numbers found (unsolved problem)

Special Number Types in Sequences

• Amicable numbers: pairs that sum to each other's proper factors
  • 220 ↔ 296 (lowest amicable pair)
  • Demonstrated with physical keychains showing 220 and 296
• Aspiring numbers: eventually reach perfect number in sequence
  • 95 → eventually reaches 6 (perfect number)
  • 25 → 6 directly
• Sociable numbers: form loops of length > 2

Computational Exploration

• Used programming (initially Joji, then Python for heavy calculations) to explore sequences
• 138 sequence: dramatic journey reaching ~18 billion before collapsing to 1
  • Required logarithmic scale plotting to visualize full journey
  • Demonstrates mountain-range pattern in log scale
• Linear vs logarithmic scaling: log scale essential for viewing multi-magnitude sequences

The Lehmer Five and Unsolved Problems

• 276 (double of 138): first number with unknown sequence fate
  • Computed for 70+ steps, reaching beyond 10^13
  • Requires supercomputer resources for further exploration
• Catalan-Dickson Conjecture: do all aliquot sequences either:
  • End in 1, or
  • Hit perfect number and loop, or
  • Form amicable/sociable cycles
• Lehmer Five: five numbers <1000 with unknown sequence outcomes
  • 276, 552, 564, 660, 966
  • Some numbers (like 306) eventually merge into same sequence as 276

Additional Discoveries

• 1184 ↔ 1210: second smallest amicable pair, discovered 200 years after first pair
• Untouchable numbers: numbers that cannot be aliquot sum of any other number
  • 5 is only known odd untouchable number
  • Unsolved: is 5 the only odd untouchable number?

Mathematical Significance

• Low-hanging fruit in recreational number theory
• Connects familiar concepts (perfect, abundant, deficient) with deep unsolved problems
• Demonstrates limits of current computational power in number theory