OFFSET
1,1
COMMENTS
The integers m counted are A038770, so A038770(a(n)) = 10^n-1 is the last of n digits, and A038770(a(n)+1) = 10^n is the first of n+1 digits, for n>=1.
The digit divisibility condition is a regular language so a(n) is a linear recurrence. Working through a state machine for A038770 shows the recurrence is order 984, though its characteristic polynomial factorizes over rationals into terms of orders at most 36. The recurrence begins at a(4)..a(987) giving a(988). See the links for coefficients and generating function.
The biggest root (by magnitude) of the recurrence characteristic polynomial is 10 and its g.f. coefficient is 1 which shows a(n) -> 10^n. Or simply the number of m containing at least one digit 1 (a subset of those counted here) approaches 10^n per A016189.
LINKS
FORMULA
a(n) = 10^n-1 - A327561(n).
MATHEMATICA
Accumulate@ Array[Count[Range[10^(# - 1), 10^# - 1], _?(MemberQ[Divisible[#, Cases[IntegerDigits[#], Except[0]]], True] &)] &, 7] (* Michael De Vlieger, Sep 30 2019, after Harvey P. Dale at A038770 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Kevin Ryde, Sep 16 2019
STATUS
approved