%I #28 Aug 22 2021 13:30:11

%S 9,68,708,7578,79952,832506,8585583,87944417,896452992,9104962748,

%T 92222435013,932113080563,9405187219507,94771322677210,

%U 953907792350911,9592689086414459,96392955785210896,967997194404428275,9715595409791983073

%N The number of integers m in the range 0 < m < 10^n which are divisible by one or more of their own digits (A038770).

%C 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.

%C 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.

%C 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.

%H Kevin Ryde, <a href="/A327560/b327560.txt">Table of n, a(n) for n = 1..999</a>

%H Kevin Ryde, <a href="/A327560/a327560.gp.txt">Linear recurrence coefficients and generating function, in a PARI/GP script</a>

%H Kevin Ryde, <a href="/A327560/a327560.pl.txt">Perl program making a state machine with Foma to verify the linear recurrence</a>

%H <a href="/index/Rec#order_984">Index entries for linear recurrences with constant coefficients</a>, order 984.

%F a(n) = 10^n-1 - A327561(n).

%t 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 *)

%Y Cf. A038770 (digit strings), A327561 (opposite counts).

%K nonn,base

%O 1,1

%A _Kevin Ryde_, Sep 16 2019