1, 2, 3, 4, 5, 6, 9, and 10.
Money can’t buy you back the love that you had then.
—Feist
Twenty-four years ago I wrote in the New York Times about Neil J. A. Sloane, a Bell Labs mathematician who had become, single-handed, a unique information-age resource. He was the world’s archivist of number sequences. If you had a question in the form What comes next after 1, 3, 9, 26, 73, 194? Sloane was your go-to guy.
And he still is. When I met him in 1987, his classic book, A Handbook of Integer Sequences, was already outdated; he was planning a new edition, and he had thousands of sequences lying about in files and cartons. (Lest we forget, children, the word files implied paper.) Now, of course, it is online. It has recently moved from Sloane’s personal web page to a home of its own, supported by a new OEIS Foundation, tax-exempt and self-sustaining, meant to maintain the On-Line Encyclopedia of Integer Sequences™ “indefinitely.” Still, it’s a little hand-to-mouth, and Sloane worries about the long term.
You can still buy the Handbook used; it contains a hefty 2,372 sequences. The online version contains 184,846, as of today. It is, of course, searchable. It offers puzzles, plots, an index, and a wiki.
Many sequences have formal names (doubly automorphic primes; subfactorial or rencontres numbers; Hexanacci numbers) and profound mathematical significance. Others, not so much. There are zig numbers and zag numbers; unhappy numbers and untouchable numbers. Here’s a silly-looking one, labeled A003459: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 … These are “permutable primes”: prime numbers that remain prime no matter how you rearrange their digits.
Sequences can be based on dice throws, chess games, pie slices, beads on necklaces, and lately, even mass transit. I wrote in 1987 that Sloane did not permit sequences like 14, 18, 23, 28, 34, 42, 50, 59 . . . —the local stops on the West Side IRT. That omission has now been rectified; it is sequence A000053.
Most sequences grow and grow; others get into loops; still others don’t seem to go anywhere at all: 0, 1, 2, 1, 1, 2, 2, 1, 1, 0, 3, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 3, 1, 0, 1, 3, 2, 2, 2, 1, 3, 2, 0, 2, 1, 1, 4, 2, 1, 3, 2, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 3, 3, 1, 3, 2, 0 … (“number of partitions of n into a prime and a square.”) Some sequences grow so fast they can barely be listed. The so-called Busy Beaver function, defined as “the maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting,” starts 1, 6, 21, 107, and then—oops—no one even knows. There must be an answer for the fifth, sixth, and subsequent terms, because the function is well defined; however, it is also noncomputable. (If that seems paradoxical, we have Alan and Kurt
to thank, as usual.) The fifth Busy Beaver number is known to be greater than 47 million, and the sixth is known to be greater than 1 followed by 1,700 zeroes. Scott Aaronson explains nicely in his essay, Who Can Name the Bigger Number? and we’re definitely in Gregory Chaitin territory here.
The OEIS is a living organism now, more and more in the hands of its users, wiki-style. There are deleted sequences (dozens every month) and sequences awaiting review. I asked Sloane if he had any current favorites and he mentioned this “toothpick sequence”: 0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 79, 95, 123, 155. It corresponds to a fractal two-dimensional cellular automaton—more fun to watch than to explain:
