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

[Allan Wechsler]

I recommend that we back this sequence up with a more fundamental one,
DD(n), the number of different digits that can be achieved by playing the
SMI game starting with n. The sequence we are discussing is "n such that
DD(n) = 10".

DD(n) can take on any value from 1 to 10, inclusive. There are lots of
obvious questions about it. Clearly (by induction), DD(10k) = 1, but are
there arbitrarily large n with DD(n) = 1 that are *not *multiples of 10? If
not, what is the largest such n?

On Mon, Apr 3, 2023 at 9:36 PM Tim Peters <tim.peters at gmail.com> wrote:

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