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A294496 Number of distinct minimal period lengths of periodic infinite words on n symbols having the constant gap property.

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%I A294496 #17 Nov 17 2017 08:45:49
%S A294496 1,1,2,3,6,8,15,18,31,35,56,62
%N A294496 Number of distinct minimal period lengths of periodic infinite words on n symbols having the constant gap property.
%C A294496 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.
%C A294496 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.
%H A294496 Jeffrey Shallit, <a href="/A294496/a294496.txt">list of distinct moduli for disjoint covering systems</a>
%H A294496 I. P. Goulden, L. B. Richmond, and J. Shallit, <a href="https://arxiv.org/abs/1711.04109">Disjoint covering systems and the reversion of the Mobius series</a>, arxiv Preprint arXiv:1711.04109 [math.NT], November 11 2017.
%e A294496 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.
%Y A294496 Cf. A050385.
%K A294496 nonn,more
%O A294496 1,3
%A A294496 _Jeffrey Shallit_, Nov 01 2017