2017
DOI: 10.48550/arxiv.1705.01883
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Ulam Sequences and Ulam Sets

Noah Kravitz,
Stefan Steinerberger

Abstract: The Ulam sequence is given by a 1 = 1, a 2 = 2, and then, for n ≥ 3, the element an is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence 1, 2, 3, 4, 6, 8, 11, 13, 16, . . . , which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors {v 1 , . . . , v k } ⊂ R n , we define a sequence by repeatedly adding the smallest elements t… Show more

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