cp's OEIS Frontend

A327560 The number of integers m in the range 0 < m < 10^n which are divisible by one or more of their own digits (A038770).

%I A327560 #28 Aug 22 2021 13:30:11
%S A327560 9,68,708,7578,79952,832506,8585583,87944417,896452992,9104962748,
%T A327560 92222435013,932113080563,9405187219507,94771322677210,
%U A327560 953907792350911,9592689086414459,96392955785210896,967997194404428275,9715595409791983073
%N A327560 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 A327560 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 A327560 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 A327560 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 A327560 Kevin Ryde, <a href="/A327560/b327560.txt">Table of n, a(n) for n = 1..999</a>
%H A327560 Kevin Ryde, <a href="/A327560/a327560.gp.txt">Linear recurrence coefficients and generating function, in a PARI/GP script</a>
%H A327560 Kevin Ryde, <a href="/A327560/a327560.pl.txt">Perl program making a state machine with Foma to verify the linear recurrence</a>
%H A327560 <a href="/index/Rec#order_984">Index entries for linear recurrences with constant coefficients</a>, order 984.
%F A327560 a(n) = 10^n-1 - A327561(n).
%t A327560 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 A327560 Cf. A038770 (digit strings), A327561 (opposite counts).
%K A327560 nonn,base
%O A327560 1,1
%A A327560 _Kevin Ryde_, Sep 16 2019