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

Wed Apr 5 03:32:19 CEST 2023

Here are the terms that I have for the newly created A361340. The ?'s are
terms which I am currently unable to calculate (I know the value is at
least greater than or equal to 2^30). It would be great to have someone
else independently confirm these terms and maybe fill in some of the
unknowns.

15, 23, 119, 167, 12049, 424, 735, 907, 17117, 1250, 307747, 2703, 49225,
9422, 57823, 5437, 2076131, 7747, 639987, 44960, 822799, 11537, 23809465,
24967, 1539917, 109346, 4643181, 26357, ?, 37440, 1885949, 285085, 7782015,
265806, ?, 66524, 8340541, 699890, 158607997, 85684, ?, 97734, 34060385,
5703971, 33087047, 121325, ?, 224759, 471244801, 3159907, 116984813,
165147, ?, 831511, 644158715, 2578087, 76065557, 306313, ?, 292432,
186679881, 17598289.

All of these unknowns are cases where the base is a multiple of 6 which
appears to grow much faster than the surrounding terms (over this short
window at least).

As an additional observation, it appears that prime number bases reach all
single-digits faster than any other. I wonder if there is a nice
explanation for this.

- Zach

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.
> >> >
> >> > --
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> >> >
> >>
> >> --
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> >>
> >
>
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