[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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