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

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

Author

Steve Butler, Jan 20 2015

Keywords

• nonn
• base

Comments

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.

Examples

			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.)
		

Links

• 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.

Crossrefs

Cf. A253057, A253058, A253953.