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

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, 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, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset

0,10

Comments

The total number of digits >= 8 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Links

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Formula

From Hieronymus Fischer, Jun 10 2012: (Start)

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)).

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)).

a(10^m-1) = 2*m*10^(m-1). (this is total number of digits >= 8 occurring in all the numbers with <= m places).

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)

Maple

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

Crossrefs

Partial sums of A102681.

Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.

Cf. A000120, A000788, A023416, A059015 (for base 2).

Sequence in context: A167661 A187187 A300358 * A116371 A103377 A343336

Adjacent sequences: A102679 A102680 A102681 * A102683 A102684 A102685

Keyword

nonn,base,easy

Author

N. J. A. Sloane, Feb 03 2005

Extensions

More terms from Emeric Deutsch, Feb 23 2005

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

Status

approved