cp's OEIS Frontend

A102682 Number of digits >= 8 in the decimal representations of all integers from 0 to n.

%I A102682 #24 Nov 16 2022 17:27:36
%S A102682 0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,
%T A102682 6,6,6,6,7,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,10,10,10,10,11,12,12,12,
%U A102682 12,12,12,12,12,12,13,14,14,14,14,14,14,14,14,14,15,16,17,18,19,20,21,22
%N A102682 Number of digits >= 8 in the decimal representations of all integers from 0 to n.
%C A102682 The total number of digits >= 8 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - _Hieronymus Fischer_, Jun 10 2012
%H A102682 Hieronymus Fischer, <a href="/A102682/b102682.txt">Table of n, a(n) for n = 0..10000</a>
%F A102682 From _Hieronymus Fischer_, Jun 10 2012: (Start)
%F A102682 a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/5)*(2n + 2 - (3/5 + floor(n/10^j + 1/5))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
%F A102682 a(n) = (n+1)*A102681(n) + (1/2)*Sum_{j=1..m+1} ((-3/5*floor(n/10^j + 1/5) + floor(n/10^j))*10^j - (floor(n/10^j + 1/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
%F A102682 a(10^m-1) = 2*m*10^(m-1). (this is total number of digits >= 8 occurring in all the numbers with <= m places).
%F A102682 G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(8*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)
%p A102682 p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=8 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..95); # _Emeric Deutsch_, Feb 23 2005
%Y A102682 Partial sums of A102681.
%Y A102682 Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
%Y A102682 Cf. A000120, A000788, A023416, A059015 (for base 2).
%K A102682 nonn,base,easy
%O A102682 0,10
%A A102682 _N. J. A. Sloane_, Feb 03 2005
%E A102682 More terms from _Emeric Deutsch_, Feb 23 2005
%E A102682 An incorrect g.f. was deleted by _N. J. A. Sloane_, Sep 16 2009