A253953 -id:A253953 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 13 2015

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

			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.

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

Cf. A253058, A253952, A253953, A293929.

A253952 Numbers that require three steps to collapse to a single digit in base 4 (written in base 10).

Original entry on oeis.org

43, 103, 139, 154, 163, 169, 223, 343, 403, 463, 523, 547, 553, 610, 643, 649, 673, 703, 823, 847, 862, 1231, 1303, 1363, 1486, 1603, 2059, 2083, 2089, 2179, 2185, 2209, 2239, 2434, 2563, 2569, 2593, 2623, 2689, 2731

Offset: 1

Steve Butler, Jan 20 2015

One step consists of taking the number in base 4 and inserting some plus signs between the digits with no restrictions and adding the resulting numbers together in base 4. The numbers given here cannot be taken to a single digit in one or two steps. It is known that three steps always suffice to get to a single digit, and that there are infinitely many numbers that require three steps.

			As an example a(1)=43 which in base 4 can be written as 223.  There are then three ways to insert plus signs in the first step:
2+23   22+3   2+2+3
This gives the numbers (in base 4) as 31, 31, and 13 respectively.  In the second step we have one of the following two:
3+1   1+3
In both cases this gives the number (in base 4) of 10.  Finally in the third step we have the following:
1+0
Which gives 1, a single digit, and we cannot get to a single digit in one or two steps.  (Note, the single digit that we reduce to is independent of the sequence of steps taken.)

Steve Butler, Table of n, a(n) for n = 1..638
S. Butler, R. Graham and R. Stong, Partition and sum is fast, arXiv:1501.04067 [math.HO], 2014.

Cf. A253057, A253058, A253953.

cp's OEIS Frontend

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

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