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

[Tim Peters]

I believe the smallest int from which all single digits can be obtained
is 17117. Here are ways to get 0 through 9, in order:

1*7117 = 7117
7*117 = 819
8*19 = 152
1*52 = 52
5*2 = 10
1*0 = 0

1711*7 = 11977
1197*7 = 8379
837*9 = 7533
7*533 = 3731
373*1 = 373
37*3 = 111
1*11 = 11
1*1 = 1

1*7117 = 7117
711*7 = 4977
4*977 = 3908
3*908 = 2724
2*724 = 1448
14*48 = 672
67*2 = 134
1*34 = 34
3*4 = 12
1*2 = 2

1*7117 = 7117
711*7 = 4977
49*77 = 3773
377*3 = 1131
1*131 = 131
1*31 = 31
3*1 = 3

1*7117 = 7117
711*7 = 4977
4*977 = 3908
3*908 = 2724
2*724 = 1448
144*8 = 1152
11*52 = 572
57*2 = 114
1*14 = 14
1*4 = 4

1711*7 = 11977
119*77 = 9163
91*63 = 5733
573*3 = 1719
171*9 = 1539
153*9 = 1377
137*7 = 959
9*59 = 531
5*31 = 155
15*5 = 75
7*5 = 35
3*5 = 15
1*5 = 5

1*7117 = 7117
7*117 = 819
81*9 = 729
72*9 = 648
6*48 = 288
28*8 = 224
2*24 = 48
4*8 = 32
3*2 = 6

1711*7 = 11977
1197*7 = 8379
837*9 = 7533
7*533 = 3731
37*31 = 1147
11*47 = 517
51*7 = 357
3*57 = 171
1*71 = 71
7*1 = 7

1*7117 = 7117
7*117 = 819
81*9 = 729
72*9 = 648
6*48 = 288
2*88 = 176
1*76 = 76
7*6 = 42
4*2 = 8

1*7117 = 7117
711*7 = 4977
49*77 = 3773
377*3 = 1131
113*1 = 113
11*3 = 33
3*3 = 9

On Mon, Apr 3, 2023 at 3:07 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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>