HEXAGONbeta

On a Family of Counterexamples to Richard Guy's Mex Sequence Problem

Primarily human-written textHuman understanding: all partsmath.NT — Number Theory

Contributed by Shuta Iwai ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We give a family of counterexamples to Richard Guy's problem E27 with initial terms of length five adding onto the recent example of length eight. Additionally, we prove that for length one, one always has a ultimately periodic sequence. Lastly we pose a problem regarding the notion of a zth mex sequence.

Provenance statement

Claude Opus 5.5 with thinking level set to high was used to write code which computed the first one hundred values for thousands of different sequences. In particular, the sequence 2,0,2,0,2 was computed through this method as well as 2,0,3,0,2 and its families. Additionally, the final manuscript was checked using the same model for any typos or grammatical errors while the proof, generalization, and analysis were all formulated and written independently without any assistance.

Tools used

Anthropic
Claude OpusVersion 5.5

References

  1. OEIS Foundation Inc., Entry A067017 in The On-Line Encyclopedia of Integer Sequences, https://oeis.org/, (2026).link
  2. R. K. Guy, Unsolved Problems in Number Theory , Springer, New York, (2004).
  3. V. Jandhyala, A mex sequence that is ultimately not periodic , preprint, https://github.com/jvvk/mathematics/blob/main/mex-sequence-not-periodic/paper/mex.pdf, (2026).link

Version history

  1. v1Submitted by Shuta IwaiInitial depositCurrentOct 31, 2026