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

1, 4, 7, 10, 13, 19, 22, 25, 28, 31, 34, 37, 40, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 163, 166, 169, 172, 175, 178, 181, 184, 187, 190, 193, 196, 199, 202, 205, 208, 211, 214, 217, 220, 223, 226, 229, 232

Offset: 1

Author

Jeffrey Shallit, Dec 11 2020

Keywords

• nonn

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. 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.

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

Examples

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

Links

• 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.

Crossrefs

Cf. A005823, A339636.