Thursday, June 1, 2023

# What Number Comes Next? Ask the Encyclopedia of Integer Sequences.

The OEIS is long on sequences that tease and land like a joke — like the eban sequence, invented by Dr. Sloane. “It’s very, very simple,” he said. “And yet nobody ever guesses it.”

More terms are not necessarily helpful: 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, 2006, 2030 …

The punchline of the eban numbers is that the letter “e” is banned: This sequence contains no numbers that, when the numbers are spelled out, contain the letter “e.”

“Eban sounds a bit like ‘even,’ and it’s a very nice pun,” Dr. Sloane said — because if you look at numbers without an “e,” they are all even. “This is an old theorem of mine: that every odd number has an ‘e’ in it, in English. And so all the numbers where ‘e’ is banned are even. Of course, not all even numbers are eban, but enough are to make a good sequence.”

Here are a few more sequences to try (answers below):

Dr. Sloane first went looking for a sequence in 1964, as a graduate student. Studying paths in an artificial neural network (a mathematical system mimicking the human brain), his computations generated:

“I badly needed a formula for the nth term in order to determine the rate of growth of the terms,” he wrote in a retrospective published in April. “This would indicate how long the activity in this very simple neural network would persist.” Hunting through textbooks, reference books, journals, he came close, but no sequence. Eventually, together with the combinatorialist John Riordan, he figured out the formula, and the next term: 237432.

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