a sequence problem

John Jones

unread,

Sep 22, 2019, 1:47:45 PM9/22/19

to

Hello recreationalists!

I gave this ng up years ago when it appeared totally moribund.
It seems to have a couple of recent posts, so I am emboldened to add
one.
(At least it hasn't gone the way of sci.math's lunatic fringe.)

I have been pondering over a simple sequence problem, suggested at
http://www.math.utah.edu/~cherk/puzzles.html

Let a[n+1] = a[n] + prodd(n) where prodd(n) is the product of digits in
a[n].
Is there an a[1] such that the sequence is unbounded?

Immediate reaction: no - prodd looks pretty random, and basically you
are taking a random walk on a minefield of integers many of which have a
zero digit (when the sequence stops).

There are some long sequence lengths however, defined as (i-1) where
a[i] has a zero digit (so length can be zero in this definition, maybe 1
would be the better minimum).

The humble 187 has a sequence length of 27.

The maximum sequence length up to 5,000,000 is 32, at 3515987.

If the process were truly random, the probability of beginning with a 7
digit number and landing on another safe 7-digit number would be
0.538084 so it would be like always betting on red in roulette, and
expecting a safe run of over twenty bets! (IIRC this has been done.)

To get around 27 in a run you would have a chance of 5.4E-08 and an
expected number of 7 digit numbers of 0.49, which is to say 1 if you are
lucky.

In fact there are 467 just between 1000000 and 5000000 (where my
patience ran out).

Ergo, it is not even *close* to being random. But as Erdos might have
said, maybe mathematics is not ready for such problems.

And as Hardy did say, words to the effect that, there is no deep
mathematics involving decimal representations.

Anyway, there you are. Any comments?
HTH
JJ

Barry Schwarz

unread,

Sep 28, 2019, 1:24:10 PM9/28/19

to

41,863,638,649 has length 39. So does 1,141,863,638,649. There may
be others since my program normally does not report duplicates.

So far, values up to 1.32e12 have not exceeded this.

On Sun, 22 Sep 2019 18:47:28 +0100, John Jones <a1...@hotamil.com>
wrote:

--
Remove del for email

John Jones

unread,

Sep 29, 2019, 4:32:11 AM9/29/19

to

On Sat, 28 Sep 2019 10:24:10 -0700, "Barry Schwarz" <schw...@dqel.com>
wrote in article <dd5voeh6qm5hgjgea...@4ax.com>...

>
> 41,863,638,649 has length 39. So does 1,141,863,638,649. There may
> be others since my program normally does not report duplicates.
>
> So far, values up to 1.32e12 have not exceeded this.
>
> On Sun, 22 Sep 2019 18:47:28 +0100, John Jones <a1...@hotamil.com>
> wrote:
>

Well done.
Tacking 1 on the front of a number obviously often has the same sequence
length.
I have also noticed that many longer run lengths have products ending
with lots of zero digits towards the end, but haven't been able to
conclude anything from that.
Cheers
JJ

Barry Schwarz

unread,

Sep 29, 2019, 10:03:08 AM9/29/19

to

On Sun, 29 Sep 2019 09:31:50 +0100, John Jones <a1...@hotamil.com>

wrote:

>On Sat, 28 Sep 2019 10:24:10 -0700, "Barry Schwarz" <schw...@dqel.com>
>wrote in article <dd5voeh6qm5hgjgea...@4ax.com>...
>>
>> 41,863,638,649 has length 39. So does 1,141,863,638,649. There may
>> be others since my program normally does not report duplicates.
>>
>> So far, values up to 1.32e12 have not exceeded this.
>>
>> On Sun, 22 Sep 2019 18:47:28 +0100, John Jones <a1...@hotamil.com>
>> wrote:
>>
>Well done.
>Tacking 1 on the front of a number obviously often has the same sequence
>length.

Not so.

Let a[1] = 88. The sequence is 152, 162, 174, 202 with a length of 4.

Let a[1] = 188. The sequence is 252, 272, 300 with a length of 3.

Tacking 1 in front xy...z to produce 1xy...z produces the same
sequence length only if adding to xy...z never produces a carry.

John Jones

unread,

Sep 30, 2019, 3:13:09 AM9/30/19

to

On Sun, 29 Sep 2019 07:03:04 -0700, "Barry Schwarz" <schw...@dqel.com>
wrote in article <ufd1pede74o10r74m...@4ax.com>...

You are correct.
Only if there is no carry.
eg 1386776 and 11386776 both have the run length of 27.
As does 111386776 1111386776 etc.
HTH
JJ

Hen Hanna

unread,

Aug 29, 2022, 2:52:19 PM8/29/22

to

On Saturday, September 28, 2019 at 10:24:10 AM UTC-7, Barry Schwarz wrote:
> 41,863,638,649 has length 39. So does 1,141,863,638,649. There may
> be others since my program normally does not report duplicates.
>
> So far, values up to 1.32e12 have not exceeded this.
>

> On Sun, 22 Sep 2019 18:47:28 +0100, John Jones <a1...@hotamil.com>
> wrote:
> >Hello recreationalists!
> >
> >I gave this ng up years ago when it appeared totally moribund.
> >It seems to have a couple of recent posts, so I am emboldened to add
> >one.
> >(At least it hasn't gone the way of sci.math's lunatic fringe.)
> >
> >I have been pondering over a simple sequence problem, suggested at
> >http://www.math.utah.edu/~cherk/puzzles.html
> >
> >Let a[n+1] = a[n] + prodd(n) where prodd(n) is the product of digits in
> >a[n].
> >Is there an a[1] such that the sequence is unbounded?

one thought i had was... Avoiding (the digit) 0 is even harder in base 2, 3, ...

did the GOAT's (Gauss, Euler, ...) think about problems like this ? (Collatz)

miss Rhoda

unread,

Feb 21, 2023, 5:12:10 PM2/21/23

to