# [seqfan] Re: W and M numbers (was: Any digit of F is the sum of two other digits of F)

Charles Greathouse charles.greathouse at case.edu
Wed Jan 4 18:39:31 CET 2012

```So this sequence is 10-automatic: it can be recognized by a regular
expression when written in decimal. I don't think it's feasible to write it
out directly, though. But you can see that for every digit there is a
collection of 1-5 multisets of digits, at least one of which needs to be
contained in any number using the digit. For example, a number using the
digit 2 must have either 00022 or 000112. For each of the 1023 combinations
of digits, take the maximum of each involved digit to find the new multiset
for that combination. From here it's obvious that a regular expression
exists, as well as that it must be large since each order of digits in each
multiset must be encoded.

It's a pity, though, since an explicit finite automaton (from the regex)
would give a closed-form expression for the number of d-digit members. It's

Charles

On Tuesday, January 3, 2012, Eric Angelini <Eric.Angelini at kntv.be> wrote:
> Hello SeqFans,
>
> A "W number" is an integer W having the property that every digit of W is
the sum of two or more digits of W.
> Example:
> 10070231 is a W number as:
> - each digit "1" is the sum of another "1" and a "0";
> - each digit "0" is the sum of the two other "0"s;
> - the digit "7" is the sum of the digits 1+2+3+1;
> - the digit "2" is the sum of the digits 1+1;
> - the digit "3" is the sum of the digits 2+1.
> Sequence:
> The W(n) sequence of "W numbers" starts like this:
> W(n) = 10001, 10010, 10100, 11000, 20002, 20020, 20200, 22000, 30003,
30030, 30300, 33000, 40004, 40040, 40400, 44000, 50005, 50050, 50500,
55000, 60006, 60060, 60600, 66000, 70007, 70070, 70700, 77000, 80008,
80080, 80800, 88000, 90009, 90090, 90900, 99000, 100001, 100010, 100011,
100012, 100021, 100100, 100101, 100102, 100110, 100120, ...
>
> (...)
>
> "M numbers" (M for Multiply) are here:
>
> Best,
> É.
>
>
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Charles Greathouse
Analyst/Programmer
Case Western Reserve University

```