A059015 -id:A059015 - cp's OEIS frontend

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

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 5 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

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.

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

Partial sums of A102675.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
Cf. A000120, A000788, A023416, A059015 (for base 2).

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

Accumulate[Table[Total[Take[DigitCount[n],{5,9}]],{n,0,80}]] (* Harvey P. Dale, Apr 27 2015 *)

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)

More terms from Emeric Deutsch, Feb 23 2005

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

Original entry on oeis.org

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

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 6 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

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

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).

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

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)

More terms from Emeric Deutsch, Feb 23 2005

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

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

Original entry on oeis.org

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, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36

Offset: 0

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 7 occurring in all the numbers 0, 1, 2, ..., n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

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

Partial sums of A102679.
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).

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:
seq(add(p(i),i=0..n), n=0..90);
# Emeric Deutsch

Accumulate[Table[Count[IntegerDigits[n],?(#>6&)],{n,0,90}]] (* _Harvey P. Dale, Sep 04 2018 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
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)).
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)).
a(10^m-1) = 3*m*10^(m-1).
(this is total number of digits >= 7 occurring in all the numbers with <= m places).
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)

More terms from Emeric Deutsch, Feb 23 2005

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

Original entry on oeis.org

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

N. J. A. Sloane, Feb 03 2005

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

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

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).

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

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)

More terms from Emeric Deutsch, Feb 23 2005

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

A168160 Number of 0's in the matrix whose lines are the binary expansion of the numbers 1,...,n.

Original entry on oeis.org

0, 2, 2, 7, 8, 9, 9, 19, 21, 23, 24, 26, 27, 28, 28, 47, 50, 53, 55, 58, 60, 62, 63, 66, 68, 70, 71, 73, 74, 75, 75, 111, 115, 119, 122, 126, 129, 132, 134, 138, 141, 144, 146, 149, 151, 153, 154, 158, 161, 164, 166, 169, 171, 173, 174, 177, 179, 181, 182, 184, 185, 186

Offset: 1

M. F. Hasler, Nov 22 2009

The matrix is to be taken of minimal size, i.e., have n lines and the number of columns needed to write n in base 2 in the last line, A070939(n). Otherwise said, there is no zero column.
The number of zeros in the last line of the matrix is given by A023416(n).
One has a(n)=a(n-1) iff n = 2^k-1 for some k.

			a(4)=7 is the number of zeros in the matrix
[001] /* = 1 in binary */
[010] /* = 2 in binary */
[011] /* = 3 in binary */
[100] /* = 4 in binary */

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000788, A070939, A023416, A059015, A380230.

#*BitLength[#] - Accumulate[DigitCount[#, 2, 1]] & [Range[100]] (* Paolo Xausa, Jan 17 2025 *)

A168160(n)=n*#binary(n)-sum(i=1,n,norml2(binary(i)))

def A168160(n): return n*(a:=n.bit_length())-(n+1)*n.bit_count()-(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,a+1))>>1) # Chai Wah Wu, Nov 11 2024

a(n) = n*A070939(n) - A000788(n) = A380230(n) - A000788(n).

A301896 a(n) = product of total number of 0's and total number of 1's in binary expansions of 0, ..., n.

Original entry on oeis.org

0, 1, 4, 8, 20, 35, 54, 72, 117, 165, 221, 280, 352, 425, 504, 576, 726, 875, 1036, 1200, 1386, 1575, 1776, 1976, 2214, 2451, 2700, 2944, 3216, 3479, 3750, 4000, 4455, 4897, 5355, 5808, 6300, 6789, 7296, 7800, 8364, 8925, 9504, 10080, 10695, 11305, 11931, 12544, 13260, 13965, 14688

Offset: 0

Ilya Gutkovskiy, Mar 28 2018

			+---+-----+---+---+---+---+----------+
| n | bin.|0's|sum|1's|sum|   a(n)   |
+---+-----+---+---+---+---+----------+
| 0 |   0 | 1 | 1 | 0 | 0 | 1*0 =  0 |
| 1 |   1 | 0 | 1 | 1 | 1 | 1*1 =  1 |
| 2 |  10 | 1 | 2 | 1 | 2 | 2*2 =  4 |
| 3 |  11 | 0 | 2 | 2 | 4 | 2*4 =  8 |
| 4 | 100 | 2 | 4 | 1 | 5 | 4*5 = 20 |
| 5 | 101 | 1 | 5 | 2 | 7 | 5*7 = 35 |
| 6 | 110 | 1 | 6 | 2 | 9 | 6*9 = 54 |
+---+-----+---+---+---+---+----------+
bin. - n written in base 2;
0's - number of 0's in binary expansion of n;
1's - number of 1's in binary expansion of n;
sum - total number of 0's (or 1's) in binary expansions of 0, ..., n.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A000120, A000788, A023416, A059015, A071295, A301336.

b:= proc(n) option remember; `if`(n=0, [1, 0], b(n-1)+
     (l-> [add(1-i, i=l), add(i, i=l)])(Bits[Split](n)))
    end:
a:= n-> (l-> l[1]*l[2])(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2023

Accumulate[DigitCount[Range[0, 50], 2, 0]] Accumulate[DigitCount[Range[0, 50], 2, 1]]

def A301896(n): return (2+(n+1)*(m:=(n+1).bit_length())-(1<Chai Wah Wu, Mar 01 2023

def A301896(n): return (a:=(n+1)*n.bit_count()+(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1))*(2+(n+1)*(t:=(n+1).bit_length())-(1<Chai Wah Wu, Nov 11 2024

a(n) = A059015(n)*A000788(n).

a(2^k-1) = 2^(k-2)*(2^k*(k - 2) + 4)*k.

A173209 Partial sums of A000069.

Original entry on oeis.org

1, 3, 7, 14, 22, 33, 46, 60, 76, 95, 116, 138, 163, 189, 217, 248, 280, 315, 352, 390, 431, 473, 517, 564, 613, 663, 715, 770, 826, 885, 946, 1008, 1072, 1139, 1208, 1278, 1351, 1425, 1501, 1580, 1661, 1743, 1827, 1914, 2002, 2093, 2186, 2280, 2377, 2475, 2575

Offset: 1

Jonathan Vos Post, Feb 12 2010

Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.

			a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.

Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.

Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* Michael De Vlieger, Nov 01 2022 *)

a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ Charles R Greathouse IV, Mar 22 2013

a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.

a(n) = n^2 - n/2 + O(1). - Charles R Greathouse IV, Mar 22 2013

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A102676 Number of digits >= 5 in the decimal representations of all integers from 0 to n.

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

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

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

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A168160 Number of 0's in the matrix whose lines are the binary expansion of the numbers 1,...,n.

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A301896 a(n) = product of total number of 0's and total number of 1's in binary expansions of 0, ..., n.

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A173209 Partial sums of A000069.

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