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

[Neil Sloane]

I have just created three sequences: first, A361338, for the number of
different single-digit numbers that can be obtained from n.  This was
explicitly suggested by Allan W., but I gave equal credit  to Eric, since
he created the problem.
Then from Michael Branicky, the finite sequence A361339, giving the
smallest number for which A361338 is n, for n from 1 to 10,
and from Zach DeS.,the smallest number that reaches all single-digit
numbers in bases n >= 2.

Several people found that the smallest number that gives all 10 digits in
base 10 is 17117.  But what is the sequence of n such that A361338(n) = 10
? It starts 17117.  Perhaps someone could create it. And, as also suggested
by Allan, what are the numbers n such that A361338(n) = 1 ?  Can these
numbers be characterized?  I'm not sure about the n for which A361338 is 2,
3, ..., 9.   It will depend on how interesting-looking they are!

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Mon, Apr 3, 2023 at 9:53 PM Allan Wechsler <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> 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.
>> >
>> > --
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>> >
>>
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