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, 37884151, 316980400, 329892001, 329893621, 330023221, 330024841, 330039421, 331204201, 331205821, 331220401, 331958485, 344321272

Offset: 1

Author

Jeffrey Shallit, Dec 11 2020

Keywords

• nonn
• more

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.

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

Cf. A005823, A339637.

Extensions

a(11)-a(22) computed by Robert Dougherty-Bliss added by Jeffrey Shallit, Mar 11 2025