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

[Allan Wechsler]

Since Neil's message of about an hour ago, a couple of Michaels (De Vlieger
and Branicky) wrote programs in Mathematica and Python (respectively) and
produced a b-file that goes a bit past 17117. It outstripped my intuition
about what would happen. I had thought we would see a gradual shift from
small values to large ones, based on the intuition that eventually,
arbitrarily-chosen big integers would be overwhelmingly likely to reach all
10 digits. The graph shows *something *like that, but with several
surprises!

1. Values 1 *and 2* show no signs of getting less popular. Do they have an
asymptotic density greater than 0?
2. Values 3 and 4 fade surprisingly fast. Could there be a finite number of
integers that produce these values?
3. Values 5 through 10 display the pattern I expected to see, but 5 comes
on very strong, faster than I expected. Also, none of these higher values
show any detectable "fade".

Also, we get to see the next two numbers after 17117 that can achieve all
ten digits. These are 17727 and 17749.

Very cool! Thank you!



On Tue, Apr 4, 2023 at 6:03 PM Neil Sloane <njasloane at gmail.com> wrote:

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