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Week 10 · OEISOpen
Primary source

Recamán's Sequence — Does Every Integer Appear?

Simple rule · open problem· Posed 1991A005132

Problem

Start with a(0)=0a(0)=0. At step nn, first try subtracting nn from the previous term. Use the result if it is nonnegative and has not appeared before; otherwise add nn.

One-line rule: subtract when the result is allowed and new; otherwise add.

For example, 1 cannot be subtracted from 0, so the first move adds 1. Then 2 cannot be subtracted from 1, so the next move adds 2. The sequence begins 0,1,3,6,2,7,13,20,12,21,11,22,10,…0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, \ldots

Does every nonnegative integer eventually appear?

Why it matters

Bernardo Recamán Santos sent the sequence to N. J. A. Sloane in 1991; it is now OEIS A005132. The rule—subtract when allowed, otherwise add—is short, yet predicting when a value appears is difficult. A 2018 Numberphile video popularized the visualization that joins successive terms with semicircular arcs. The central question is whether the rule misses any nonnegative integer forever.

Progress so far

The OEIS computation record dated 8 February 2026 reaches term 1061210^{612}, with 852,655852{,}655 still the smallest integer not yet seen. That does not show that the number is absent forever. Any finite computation establishes only “not seen yet”; no proof is known that every integer eventually appears or that a particular integer never does.

Further reading

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

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