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

Wed Apr 5 04:36:30 CEST 2023

I put up a chart at A361340 for bases n=2..12 and terms m = 1..1000. (They look like stellar spectra.) The patterns are pretty interesting. We can create a graph of the partial sums of cardinalities of digits d of base n, that is pretty interesting, but these charts achieve some of that data in a more visually effective manner, through densities.

(The practical limit for these sort of charts is about 16384 pixels.)

Some cursory thoughts.

Prime numbers n = p have all k < p coprime to p, therefore, k * m mod p has a tendency to distribute over all digits of base p. In a base like n = 12, we have a preference for digits d such that rad(k) | rad(n).

In base 12, 3*k and 9*k produce {0, 3, 6, 9} mod 12, 4*k and 8*k produce {0, 4, 8} mod 12, and 6*k produces {0, 6} mod 12, which is why these digits appear most often. (The multiplication table of base 12 is famously patterned).

In decimal, we have 2*k, 4*k, 6*k, and 8*k producing {0, 2, 4, 6, 8} mod 10 and 5*k producing {0, 5} mod 10.

The frequencies of digits in the multiplication table of base n strongly affect the frequencies of digits seen in these studies.

I imagine once we get to very large bases (60, 120, for example) though there are a lot of patterns similar to base 12, there are also a lot of numbers coprime to 60 that walk all over every digit, so the effect might seem to be muted as n increases.

Best regards,
Mike

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of Zach DeStefano
Sent: Tuesday, 4 April, 2023 20:32
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Another Split & Multiply sequence from Eric A.

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

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