Katydid sequence

The Katydid sequence is a sequence of numbers first defined in Clifford A. Pickover's book Wonders of Numbers (2001).

Description

It is the smallest sequence of integers that can be reached from 1 by a sequence of the two operations n ↦ 2n + 2 and 7n + 7 (in any order).[1] For instance, applying the first operation to 1 produces the number 4, and applying the second operation to 4 produces the number 35, both of which are in the sequence.

The first 10 elements of the sequence are:[2]

1, 4, 10, 14, 22, 30, 35, 46, 62, 72.

Repetitions

Pickover asked whether there exist numbers that can be reached by more than one sequence of operations.[1] The answer is yes. For instance, 1814526 can be reached by the two sequences 1 – 4 – 10 – 22 – 46 – 329 – 660 – 4627 – 9256 – 18514 – 37030 – 259217 – 1814526 and 1 – 14 – 30 – 62 – 441 – 884 – 1770 – 3542 – 7086 – 14174 – 28350 – 56702 – 113406 – 226814 – 453630 – 907262 – 1814526

References
  1. Pickover, Clifford A. (2001). Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford University Press. p. 330. ISBN 9780195348002.
  2. Sloane, N. J. A. (ed.). "Sequence A060031 (Katydid sequence: closed under n -> 2n + 2 and 7n + 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.


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