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A339637 Numbers congruent to 1 (mod 3) that are the quotient of two Cantor numbers (A005823). 1


%S 1,4,7,10,13,19,22,25,28,31,34,37,40,55,58,61,64,67,70,73,76,79,82,85,

%T 88,91,94,97,100,103,106,109,112,115,118,121,163,166,169,172,175,178,

%U 181,184,187,190,193,196,199,202,205,208,211,214,217,220,223,226,229,232

%N Numbers congruent to 1 (mod 3) that are the quotient of two Cantor numbers (A005823).

%C Let C be the Cantor numbers (A005823), and let A be the set of integers congruent to 1 (mod 3) representable as the quotient of two nonzero elements of C. It is easy to see that if (3/2)*3^i < n < 2*3^i for some i, then n cannot be in A. Initial empirical data suggested that these are the only integers congruent to 1 (mod 3) not in A. However, there are additional "sporadic" counterexamples enumerated by A339636, whose structure is not well understood.

%C A simple automaton-based (or breadth-first search) algorithm can establish in O(n) time whether n is in A or not.

%H J. S. Athreya, B. Reznick, and J. T. Tyson, <a href="https://doi.org/10.1080/00029890.2019.1528121">Cantor set arithmetic</a>, Amer. Math. Monthly 126 (2019), 4-17.

%H James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, <a href="https://doi.org/10.48550/arXiv.2202.13694">Quotients of Palindromic and Antipalindromic Numbers</a>, arXiv:2202.13694 [math.NT], 2022.

%e 106 is in the sequence, because 106=1462376/13796, and 1462376 in base 3 is 2202022000002, and 13796 in base 3 is 200220222.

%Y Cf. A005823, A339636.

%K nonn

%O 1,2

%A _Jeffrey Shallit_, Dec 11 2020

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Last modified September 26 05:53 EDT 2022. Contains 356986 sequences. (Running on oeis4.)