A102676 Number of digits >= 5 in the decimal representations of all integers from 0 to n.
0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 6, 7, 8, 9, 10, 10, 10, 10, 10, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 16, 17, 18, 19, 20, 20, 20, 20, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 56, 57, 58, 59, 60, 62
Offset: 0
References
- Curtis Cooper, Number of large digits in the positive integers not exceeding n, Abstracts Amer. Math. Soc., 25 (No. 1, 2004), p. 38, Abstract 993-11-964.
Links
- Hieronymus Fischer, Table of n, a(n) for n = 0..10000
Crossrefs
Programs
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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]>=5 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..83); # Emeric Deutsch, Feb 23 2005
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Mathematica
Accumulate[Table[Total[Take[DigitCount[n],{5,9}]],{n,0,80}]] (* Harvey P. Dale, Apr 27 2015 *)
Formula
From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/2)*(2n + 2 - floor(n/10^j + 1/2)*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)*A102675(n) + (1/2)*Sum_{j=1..m+1} (floor(n/10^j)*10^j - (floor(n/10^j + 1/2)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = 5*m*10^(m-1).
(This is the total number of digits >= 5 occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(5*10^j) - x^(10*10^j))/(1-x^10^(j+1)).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} x^(5*10^j)/(1+x^(5*10^j)). (End)
Extensions
More terms from Emeric Deutsch, Feb 23 2005
Comments