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A102680 Number of digits >= 7 in the decimal representations of all integers from 0 to n.

%I A102680 #23 Nov 17 2022 05:28:58
%S A102680 0,0,0,0,0,0,0,1,2,3,3,3,3,3,3,3,3,4,5,6,6,6,6,6,6,6,6,7,8,9,9,9,9,9,
%T A102680 9,9,9,10,11,12,12,12,12,12,12,12,12,13,14,15,15,15,15,15,15,15,15,16,
%U A102680 17,18,18,18,18,18,18,18,18,19,20,21,22,23,24,25,26,27,28,30,32,34,35,36
%N A102680 Number of digits >= 7 in the decimal representations of all integers from 0 to n.
%C A102680 The total number of digits >= 7 occurring in all the numbers 0, 1, 2, ..., n (in decimal representation). - _Hieronymus Fischer_, Jun 10 2012
%H A102680 Hieronymus Fischer, <a href="/A102680/b102680.txt">Table of n, a(n) for n = 0..10000</a>
%F A102680 From _Hieronymus Fischer_, Jun 10 2012: (Start)
%F A102680 a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 7/10)*(2n + 2 - (2/5 + floor(n/10^j + 7/10))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m=floor(log_10(n)).
%F A102680 a(n) = (n+1)*A102679(n) + (1/2)*Sum_{j=1..m+1} (((-2/5)*floor(n/10^j + 7/10) + floor(n/10^j))*10^j - (floor(n/10^j + 7/10)^2 - floor(n/10^j)^2)*10^j), where m=floor(log_10(n)).
%F A102680 a(10^m-1) = 3*m*10^(m-1).
%F A102680 (this is total number of digits >= 7 occurring in all the numbers with <= m places).
%F A102680 G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(7*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)
%p A102680 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]>=7 then ct:=ct+1 else ct:=ct fi od: ct: end:
%p A102680 seq(add(p(i),i=0..n), n=0..90);
%p A102680 # _Emeric Deutsch_
%t A102680 Accumulate[Table[Count[IntegerDigits[n],_?(#>6&)],{n,0,90}]] (* _Harvey P. Dale_, Sep 04 2018 *)
%Y A102680 Partial sums of A102679.
%Y A102680 Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums of A102669.
%Y A102680 Cf. A000120, A000788, A023416, A059015 (for base 2).
%K A102680 nonn,base,easy
%O A102680 0,9
%A A102680 _N. J. A. Sloane_, Feb 03 2005
%E A102680 More terms from _Emeric Deutsch_, Feb 23 2005