A339636 Counterexamples to a conjecture about integers representable as the quotient of two Cantor numbers (A005823).

529, 592, 601, 616, 5368, 50281, 4072741, 4074361, 4088941, 4245688

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

Table of n, a(n) for n=1..10.

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

Cf. A005823, A339637.

Sequence in context: A205160 A205309 A045788 * A181414 A020289 A274932

Adjacent sequences: A339633 A339634 A339635 * A339637 A339638 A339639

Keyword

nonn,more

Author

Jeffrey Shallit, Dec 11 2020

Status

approved