|
%I #28 Jan 12 2024 08:41:05
%S 529,592,601,616,5368,50281,4072741,4074361,4088941,4245688
%N Counterexamples to a conjecture about integers representable as 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 (A339637). 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 "sporadic" counterexamples enumerated by this sequence entry, 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.
%C Conjecture: every number of the form 23*3^(4k+3) - 20 is not representable. In addition to numbers already in the sequence, for k = 3 this gives 330024841, which is also not representable.
%H Katie Anders, Madeline Locus Dawsey, Bruce Reznick, and Simone Sisneros-Thiry, <a href="https://arxiv.org/abs/2308.07252">Representations of integers as quotients of sums of distinct powers of three</a>, arXiv:2308.07252 [math.NT], 2023.
%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://arxiv.org/abs/2202.13694">Quotients of Palindromic and Antipalindromic Numbers</a>, arXiv:2202.13694 [math.NT], 2022.
%H James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, <a href="http://math.colgate.edu/~integers/w96/w96.pdf">Quotients of Palindromic and Antipalindromic Numbers</a>, INTEGERS 22 (2022), #A96.
%Y Cf. A005823, A339637.
%K nonn,more
%O 1,1
%A _Jeffrey Shallit_, Dec 11 2020
|
