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A339636 Counterexamples to a conjecture about integers representable as the quotient of two Cantor numbers (A005823). 1
529, 592, 601, 616, 5368, 50281, 4072741, 4074361, 4088941, 4245688 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
A simple automaton-based (or breadth-first search) algorithm can establish in O(n) time whether n is in A or not.
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.
LINKS
Katie Anders, Madeline Locus Dawsey, Bruce Reznick, and Simone Sisneros-Thiry, Representations of integers as quotients of sums of distinct powers of three, arXiv:2308.07252 [math.NT], 2023.
J. S. Athreya, B. Reznick, and J. T. Tyson, Cantor set arithmetic, Amer. Math. Monthly 126 (2019), 4-17.
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, INTEGERS 22 (2022), #A96.
CROSSREFS
Sequence in context: A205160 A205309 A045788 * A181414 A020289 A274932
KEYWORD
nonn,more
AUTHOR
Jeffrey Shallit, Dec 11 2020
STATUS
approved

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Last modified July 22 10:48 EDT 2024. Contains 374490 sequences. (Running on oeis4.)