A102678 Number of digits >= 6 in the decimal representations of all integers from 0 to n.

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20, 20, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 39, 40, 41, 42, 43, 44, 46, 48

Offset

0,8

Comments

The total number of digits >= 6 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 + 2/5)*(2n + 2 - (1/5 + floor(n/10^j + 2/5))*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)*A102677(n) + (1/2)*Sum_{j=1..m+1} ((-1/5*floor(n/10^j + 2/5) + floor(n/10^j))*10^j - (floor(n/10^j + 2/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).

a(10^m-1) = 4*m*10^(m-1).

(this is total number of digits >= 6 occurring in all the numbers with <= m places).

G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(6*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]>=6 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i), i=0..n), n=0..86); # Emeric Deutsch, Feb 23 2005

Crossrefs

Partial sums of A102677.

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

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

Sequence in context: A210964 A029135 A196933 * A029132 A195852 A122815

Adjacent sequences: A102675 A102676 A102677 * A102679 A102680 A102681

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