[seqfan] Re: Another Split & Multiply sequence from Eric A.

[Zach DeStefano]

If I am understanding the construction correctly (it is very possible that
I missed something), 1133 can also give 2 and 6
11 * 33 -> 3 * 63 -> 18 * 9 -> 1 * 62 -> 6 * 2 -> 1 * 2 -> 2
11 * 33 -> 3 * 63 -> 18 * 9 -> 16 * 2 -> 3 * 2 -> 6

There are plenty of numbers that generate all of the 10 single digits. The
first one I tried: 1112573, satisfies this:
11 * 12573 -> 13830 * 3 -> 4149 * 0 -> 0
111257 * 3 -> 3 * 33771 -> 1013 * 13 -> 1 * 3169 -> 31 * 69 -> 213 * 9 ->
191 * 7 -> 1 * 337 -> 3 * 37 -> 1 * 11 -> 1 * 1 -> 1
1 * 112573 -> 11257 * 3 -> 3377 * 1 -> 3 * 377 -> 11 * 31 -> 34 * 1 -> 3 *
4 -> 1 * 2 -> 2
1 * 112573 -> 11257 * 3 -> 3377 * 1 -> 3 * 377 -> 113 * 1 -> 1 * 13 -> 1 *
3 -> 3
1112 * 573 -> 637 * 176 -> 1 * 12112 -> 12 * 112 -> 13 * 44 -> 57 * 2 -> 1
* 14 -> 1 * 4 -> 4
1 * 112573 -> 1 * 12573 -> 125 * 73 -> 9 * 125 -> 11 * 25 -> 27 * 5 -> 1 *
35 -> 3 * 5 -> 1 * 5 -> 5
1 * 112573 -> 1 * 12573 -> 1257 * 3 -> 377 * 1 -> 3 * 77 -> 23 * 1 -> 2 * 3
-> 6
1 * 112573 -> 11257 * 3 -> 3 * 3771 -> 1 * 1313 -> 131 * 3 -> 39 * 3 -> 1 *
17 -> 1 * 7 -> 7
1 * 112573 -> 1 * 12573 -> 1257 * 3 -> 377 * 1 -> 37 * 7 -> 2 * 59 -> 1 *
18 -> 1 * 8 -> 8
1 * 112573 -> 11257 * 3 -> 3377 * 1 -> 3 * 377 -> 113 * 1 -> 11 * 3 -> 3 *
3 -> 9

The smallest positive number which satisfies this condition is 17117:
171 * 17 -> 290 * 7 -> 203 * 0 -> 0
1711 * 7 -> 1197 * 7 -> 837 * 9 -> 7 * 533 -> 373 * 1 -> 37 * 3 -> 1 * 11
-> 1 * 1 -> 1
171 * 17 -> 2 * 907 -> 1 * 814 -> 8 * 14 -> 1 * 12 -> 1 * 2 -> 2
1 * 7117 -> 711 * 7 -> 49 * 77 -> 377 * 3 -> 113 * 1 -> 1 * 13 -> 1 * 3 -> 3
171 * 17 -> 2 * 907 -> 1 * 814 -> 8 * 14 -> 11 * 2 -> 2 * 2 -> 4
1711 * 7 -> 1197 * 7 -> 837 * 9 -> 75 * 33 -> 247 * 5 -> 1 * 235 -> 23 * 5
-> 1 * 15 -> 1 * 5 -> 5
17 * 117 -> 19 * 89 -> 169 * 1 -> 16 * 9 -> 1 * 44 -> 4 * 4 -> 1 * 6 -> 6
1711 * 7 -> 1197 * 7 -> 837 * 9 -> 7 * 533 -> 37 * 31 -> 11 * 47 -> 51 * 7
-> 3 * 57 -> 17 * 1 -> 1 * 7 -> 7
17 * 117 -> 1 * 989 -> 98 * 9 -> 88 * 2 -> 1 * 76 -> 7 * 6 -> 4 * 2 -> 8
1 * 7117 -> 711 * 7 -> 49 * 77 -> 377 * 3 -> 113 * 1 -> 11 * 3 -> 3 * 3 -> 9

- Zach

On Mon, Apr 3, 2023 at 4:07 PM Neil Sloane <njasloane at gmail.com> wrote:

> Eric Angelini recently posted something that led to the creation of
> A361337.  These are the numbers which can reach 0 after a suitable series
> of split-and-multiply operations.
>
> I just stumbled across an older email from him where he asks if there are
> any numbers which can reach all of 0, 1, 2, ..., 9 by suitable sequences of
> split-and-multiply.  (See A361337 for the precise rules).
>
> I quote from Eric's email:
> Take the integer 1133
> We split 1133 into 1 and 133 for instance (inserting a star between two
> digits).
> (a star means multiply)
> We then make 1*133 = 133
> We iterate until we get a single digit.
>
> Question:
> Is there an integer that can reach any of the 10 single digits?
> With 1133 we can reach 0, 4, 7, 8 or 9:
> 1133 -> 11*33 -> 36*3 -> 10*8 -> 8*0 -> 0
> 1133 -> 11*33 -> 3*63 -> 1*89 -> 8*9 -> 7*2 -> 1*4 -> 4
> 1133 -> 113*3 -> 3*39 -> 1*17 -> 1*7 -> 7
> 1133 -> 113*3 -> 3*39 -> 11*7 -> 7*7 -> 4*9 -> 3*6 -> 1*8 -> 8
> 1133 -> 1*133 -> 1*33 -> 3*3 -> 9
>
> P.S. We don’t insert a star before a zero.
>
> I couldn’t find any such number.
>
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