The aliquot sequence is one of the most mysterious sequences in number theory. It repeatedly applies the sum-of-proper-divisors function, leading to various possible outcomes that are still not fully understood.
⚠The Catalan conjecture (that no sequence diverges to infinity) remains unproven. The longest unproven sequence starts at 276.
What Is an Aliquot Sequence?
An aliquot sequence is generated by repeatedly applying the sum-of-proper-divisors function. It has been studied since antiquity. The sequence can terminate, become periodic, or potentially grow forever (unknown).
Terminating
Ends at 0 when hitting a prime. 12->16->15->9->4->3->1->0. Most sequences terminate.
Perfect
s(n)=n, so sequence becomes constant: 6->6. Also 28->28, 496->496. The end of the sequence.
Amicable
2-cycle: 220->284->220. Also 1184->1210->1184. The sequence alternates forever.
Sociable
Cycles of length >2. The smallest has period 5: 12496->14288->15472->14536->14264->12496.
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FAQs about Aliquot Sequences
What is an aliquot sequence?▼
Repeatedly replace n with sum of proper divisors until reaching 0 or a cycle. 12;16;15;9;4;3;1;0 terminates.
What numbers end in a perfect number?▼
Numbers like 6, 25, 95, 119 reach 6 (perfect). Numbers like 28, 42, 140 reach 28. s(6)=6, so it stays constant.
What is the aliquot conjecture?▼
Catalan-Dickson conjecture: every aliquot sequence eventually terminates or becomes periodic. This is unproven. Number 276 has been tracked for over 2000 terms without resolution.
What is s(6)?▼
s(6)=1+2+3=6 (perfect number). The aliquot sequence of 6 is constant: 6,6,6... For prime numbers, s(p)=1, then s(1)=0.
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