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

[Zach DeStefano]

To follow up on this, starting with base 2, here is the start of the list
of the smallest numbers which satisfy the property that the
split-and-multiply technique in base b can produce all b single-digit
numbers:
15, 23, 119, 167, 12049, 424, 735, 907, 17117, 1250, 307747, 2703, 49225,
9422, 57823, 5437, 2076131, 7747, 639987,...

This does not appear anywhere in the OEIS, and it isn't too difficult to
calculate more terms. Would it be an interesting addition?

- Zach

On Mon, Apr 3, 2023 at 6:18 PM Zach DeStefano <zachdestefano at gmail.com>
wrote:

> 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.
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>