# Seven Staggering Sequences

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Homage to a Pied Puzzler ()
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#### Abstract

When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column that "every recreational mathematician should buy a copy forthwith." That book contained 2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains 117000 sequences. This paper will describe seven that I find especially interesting. These are the EKG sequence, Gijswijt's sequence, a numerical analog of Aronson's sequence, approximate squaring, the integrality of n-th roots of generating functions, dissections, and the kissing number problem. (Paper for conference in honor of Martin Gardner's 91st birthday.)

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# Seven Staggering Sequences -

Seven Staggering Sequences N. J. A. Sloane Algorithms and Optimization Department AT&T Shannon Lab Florham Park, NJ 07932���0971 Email address: njas@research.att.com April 3, 2006 0. Introduction When the Handbook of Integer Sequences came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column for July 1974 that ���every recreational mathematician should buy a copy forthwith.��� That book contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (or OEIS) [24] contains 117000 sequences. The following are seven that I find especially interesting. Many of them quite literally stagger. The sequences will be labeled with their numbers (such as A064413) in the OEIS. Much more information about them can be found there and in the references cited. 1. The EKG sequence (A064413, due to Jonathan Ayres). The first three sequences are defined by unusual recurrence rules. The first begins with a(1) = 1,a(2) = 2, and the rule for extending it is that the next term, a(n + 1), is taken to be the smallest positive number not already in the sequence which has a nontrivial common factor with the previous term a(n). Since a(2) = 2, a(3) must be even, and is therefore 4 a(4) must have a factor in common with 4, that is, must also be even, and so a(4) = 6. The smallest number not already in the sequence that has a common factor with 6 is 3, so a(5) = 3, and so on. The first 18 terms are 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20,... . It is clear that if a prime p appears in the sequence, 2p will be the term either immediately before or after it. Jeffrey Lagarias, Eric Rains and I studied this sequence in [18]. One of the things that we observed was that in fact every odd prime p was always preceded by 2p, and always followed by 3p. This is certainly true for the first 10000000 terms, but we were unable to prove it in general. We called this the EKG sequence, since it looks like an electrocardiogram when plotted (Figs. 1, 2). 1 arXiv:0912.2394v1 [math.CO] 12 Dec 2009
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n a(n) 0 20 40 60 80 100 0 20 40 60 80 100 120 140 Figure 1: The first 100 terms of the EKG sequence, with successive points joined by lines. n a(n) 800 850 900 950 1000 400 600 800 1000 1200 1400 1600 Figure 2: Terms 800 to 1000 of the EKG sequence. There is an elegant three-step proof that every positive number must eventually appear in the sequence. (i) If infinitely many multiples of some prime p appear in the sequence, then every multiple of p must appear. (For if not, let kp be the smallest missing multiple of p. Every number below kp either appears or it doesn���t, but once we get to a multiple of p beyond all those terms, the next term must be kp, which is a contradiction.) (ii) If every multiple of a prime p appears, then every number appears. (The proof is similar.) (iii) Every number appears. (For if there are only finitely many different primes among the prime factors of all the terms, then some prime must appear in infinitely many terms, and 2

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