A253057 Complete list of numbers that take three steps to collapse to a single digit in base 3 (written in base 10).

1781, 3239, 3887, 11177, 14821, 33047, 41065, 43981, 98657, 131461, 393901

Offset: 1

Author

N. J. A. Sloane, Jan 13 2015

Keywords

• nonn
• fini
• full
• nice

Comments

From Simon Demers, Oct 20 2017: (Start)

This an exceptionally nice finite sequence based on a surprisingly simple but nontrivial rule: collapse the number expressed in base 3 by inserting plus signs and adding, while minimizing the number of steps (applications).

Butler et al. (2014) proved that any number written in base 2 can be collapsed to a single digit in at most two steps. Any number written in base 3 can be collapsed to a single digit in at most two steps except, surprisingly, for the 11 numbers listed in this sequence. One thing that separates base 3 from larger bases is that there are only 11 base-3 numbers that require three applications!

Let m be the sum of the digits in base-3 expansion. Butler et al. (2014) showed that candidates for this sequence must have m < 82.

(End)

In base 3, the terms are written as 2102222, 11102222, 12022222, 120022222, 202022221, 1200022222, 2002022221, 2020022221, 12000022222, 20200022221, 202000022221. - Andrey Zabolotskiy, Oct 20 2017

Examples

			In base 3, one of the possible ways to collapse a(1) in three steps is as follows:
2102222 -> 2102 + 222 = 10101 -> 1 + 01 + 01 = 10 -> 1 + 0 = 1.
		

Links

• Steve Butler, R. L. Graham, and Richard Stong, Collapsing numbers in bases 2, 3, and beyond, in Gathering for Gardner 10, 2014.

Crossrefs

Cf. A253058, A253952, A253953, A293929.