[seqfan] Re: Remeven numbers

Kevin Ryde user42_kevin at yahoo.com.au
Fri Jan 10 06:53:34 CET 2020

oeis at hasler.fr (M. F. Hasler) writes:
> oeis.org/A330981 and oeis.org/A330982.

A variant easier to consider could be

    "alleven" = integers where remainders modulo 1 to 9 are all even,
                and digit 0 not allowed

A state machine for that one is 79 states and its recurrence etc for how
many digit strings of length k has biggest term 32/567 * 9^k.  But I
have yet to triple check that ...

Alleven is a subset of remeven.  Extras in remeven are when fewer than
all digits 1 to 9 occur, so at most 8 different digits, so at most 9*8^k
extras, which is smaller than power 9^k, so should have same limit
32/567 * 9^k remeven digit strings of length k.

For how many <= any integer n, I tried counting alleven where the start
state is something different.  This is some initial digit string
followed by k more digits.  Looks like limits 32/567 * 9^k no matter
what initial digits.  I think I'm persuaded this ought to mean

                   num remeven <= n
     lim    ---------------------------------   = 32/567
    n->inf  num integers without 0 digit <= n
                   (A052382, A324161)

Trying that ratio on some actual counts, alleven approaches 32/567 quite
quickly, but remeven is disconcertingly slow.  Perhaps that's to be
expected if you're waiting for a term in 9^k to squash terms in 8^k.

Can this be had an easier way?  The absence of digit 0 seems to upset
simple ways to think about remainders.  Oh, and I don't like my name
"alleven" so something better :).

In general, is there a name for a regular language where the number of
strings with a particular prefix has the same limit as number of all
strings?  I think I mean for any given fixed string "prefix"

             num strings <prefix>.<length k>
     lim     -------------------------------  = 1
    k->inf        num strings <length k>

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