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

[wnmyers wnmyers]

The original email stated "We don’t insert a star before a zero", but I assume that the last digit is an exception to allow multiplication by zero.

If a number contains a zero digit, you'd have to get rid of all of the zero digits for DD(n)>1. For example, 1035 allows you to get rid of the zero by 103*5->515, but you can't get rid of both zeroes in 10035. You can construct arbitrarily large numbers with DD(n) = 1 that contain lots of zeroes. Finding numbers where DD(n)=1 and that digit is not 0 is more challenging. 333 is example, but not a large one.

> On April 3, 2023 at 6:53 PM Allan Wechsler <acwacw at gmail.com mailto:acwacw at gmail.com > wrote:
> 
> 
>     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 mailto: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 mailto: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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> > > 
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