OFFSET
1,3
COMMENTS
A periodic infinite word consists of a block x repeated infinitely to the right: X = x^omega = xxx.... The minimal period length of such a word X is the length of the shortest word y such that X = y^omega. Such a word has the constant-gap property if for each letter i occurring in the word, there is a constant c_i such that two consecutive occurrences of i are separated by exactly c_i symbols. For example (0102)^omega is a constant-gap word on 3 symbols with minimal period length 4.
Alternatively, this is the number of distinct lcm's of moduli that can appear in a disjoint covering system of the integers consisting of n congruences. Disjoint covering systems and constant-gap periodic sequences are in 1-1 correspondence. For example, the covering system corresponding to (0102)^omega is x == 0 (mod 2), x == 1 (mod 4), x == 3 (mod 4), and the lcm of the moduli (2,4,4) is 4.
LINKS
Jeffrey Shallit, list of distinct moduli for disjoint covering systems
I. P. Goulden, L. B. Richmond, and J. Shallit, Disjoint covering systems and the reversion of the Mobius series, arxiv Preprint arXiv:1711.04109 [math.NT], November 11 2017.
EXAMPLE
For n = 3 the 3 constant gap words on 3 symbols are (0102)^omega, (0121)^omega, (012)^omega, with minimal period lengths 4,4,3, respectively, so 2 distinct period lengths.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jeffrey Shallit, Nov 01 2017
STATUS
approved