A072602 -id:A072602 - cp's OEIS frontend

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

1, -1, 0, -2, 1, -1, -1, -3, 2, 0, 0, -2, 0, -2, -2, -4, 3, 1, 1, -1, 1, -1, -1, -3, 1, -1, -1, -3, -1, -3, -3, -5, 4, 2, 2, 0, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, -2, -4, 2, 0, 0, -2, 0, -2, -2, -4, 0, -2, -2, -4, -2, -4, -4, -6, 5, 3, 3, 1, 3, 1, 1, -1, 3

Offset: 0

Clark Kimberling

-Sum_{n>=1} a(n)/((2*n)*(2*n+1)) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).

a(A072600(n)) < 0; a(A072601(n)) <= 0; a(A031443(n)) = 0; a(A072602(n)) >= 0; a(A072603(n)) > 0; a(A031444(n)) = 1; a(A031448(n)) = -1; abs(a(A089648(n))) <= 1. - Reinhard Zumkeller, Feb 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).
Index entries for sequences related to binary expansion of n

Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.
Cf. A066879, A094640, A110625, A145037.
Cf. A072600, A072601, A072602, A072603, A031444, A031448, A089648.
See A268289 for a recurrence based on this sequence.

a037861 n = a023416 n - a000120 n  -- Reinhard Zumkeller, Aug 01 2013

A037861:= proc(n) local L;
     L:= convert(n,base,2);
     numboccur(0,L) - numboccur(1,L)
end proc:
map(A037861, [$0..100]); # Robert Israel, Mar 08 2016

Table[Count[ IntegerDigits[n, 2], 0] - Count[IntegerDigits[n, 2], 1], {n, 0, 75}]

a(n) = if (n==0, 1, 1 + logint(n, 2) - 2*hammingweight(n)); \\ Michel Marcus, May 15 2020 and Jun 16 2020

def A037861(n):
    return 2*format(n,'b').count('0')-len(format(n,'b')) # Chai Wah Wu, Mar 07 2016

From Henry Bottomley, Oct 27 2000: (Start)
a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n).
a(2*n) = a(n) + 1; a(2*n + 1) = a(2*n) - 2 = a(n) - 1. (End)
G.f. satisfies A(x) = (1 + x)*A(x^2) - x*(2 + x)/(1 + x). - Franklin T. Adams-Watters, Dec 26 2006
a(n) = b(n) for n > 0 with b(0) = 0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller, Dec 31 2007
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(2^k)*(x^(2^k) - 1)/(1 + x^(2^k)). - Ilya Gutkovskiy, Apr 07 2018

A072600 Numbers which in base 2 have fewer 0's than 1's.

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 39, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115

Offset: 1

Reinhard Zumkeller, Jun 23 2002

A037861(a(n)) < 0.

b_k = {a(n) | for all n s.t. a(n) contains k binary digits equal to 1} is the list of all valid win/loss round sequences in a "best of 2k-1" two player game, where 1 is a win and 0 is a loss. For example 19 = 10011b represents a game where the winner won the first two rounds, lost the next two, and won the last one. |b_k| = A001700(k). - Philippe Beaudoin, May 14 2014

			11 is present because '1011' contains 1 '0' and 3 '1's: 1<3.

T. D. Noe, Table of n, a(n) for n = 1..4733 ( numbers < 2^13)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072601, A031443, A072602, A072603, A037861

a072600 n = a072600_list !! (n-1)
a072600_list = filter ((< 0) . a037861) [0..]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[130],DigitCount[#,2,0]Harvey P. Dale, Jan 12 2011 *)

is(n)=2*hammingweight(n)>exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

A072603 Numbers which in base 2 have more 0's than 1's.

Original entry on oeis.org

4, 8, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 140, 144, 145, 146, 148, 152, 160, 161, 162, 164, 168, 176, 192, 193

Offset: 1

Reinhard Zumkeller, Jun 23 2002

			8 is present because '1000' contains 3 '0's and 1 '1': 3>1.

T. D. Noe, Table of n, a(n) for n = 1..5202 (numbers up to 2^14)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072600, A072601, A031443, A072602.

Cf. A037861(a(n)) > 0.

a072603 n = a072603_list !! (n-1)
a072603_list = filter ((> 0) . a037861) [1..]
-- Reinhard Zumkeller, Mar 31 2015

gtQ[n_] := Module[{a, b}, {a, b} = DigitCount[n, 2]; b > a]; Select[Range[2^8], gtQ] (* T. D. Noe, Apr 20 2013 *)
Select[Range[200],DigitCount[#,2,0]>DigitCount[#,2,1]&] (* Harvey P. Dale, Feb 26 2023 *)

is(n)=2*hammingweight(n)Charles R Greathouse IV, Apr 18 2020

A072601 Numbers which in base 2 have at least as many 1's as 0's.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103

Offset: 1

Reinhard Zumkeller, Jun 23 2002

			8 = 1000_2 is not present (one '1', three '0's).
10 is present because 10=1010_2 contains 2 '0's and 2 '1's: 2<=2;
11 is present because 11=1011_2 contains 1 '0' and 3 '1's: 1<=3.

T. D. Noe, Table of n, a(n) for n = 1..5370 (numbers < 2^13)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A061854.
Cf. A037861(a(n)) <= 0.
Cf. A072600 (#0's < #1's), this seq (#0's <= #1's), A031443 (#0's = #1's).
Cf. A072602 (#0's >= #1's), A072603 (#0's > #1's), A044951 (#0's <> #1's).

a072601 n = a072601_list !! (n-1)
a072601_list = filter ((<= 0) . a037861) [0..]
-- Reinhard Zumkeller, Aug 01 2013

geQ[n_] := Module[{a, b}, {a, b} = DigitCount[n, 2]; a >= b]; Select[Range[103], geQ] (* T. D. Noe, Apr 20 2013 *)
Select[Range[110],DigitCount[#,2,1]>=DigitCount[#,2,0]&] (* Harvey P. Dale, Aug 12 2023 *)

is(n)=2*hammingweight(n)>exponent(n) \\ Charles R Greathouse IV, Apr 18 2020

A044951 Numbers having a different number of ones and zeros in base 2.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 39, 40, 43, 45, 46, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Clark Kimberling

			From _Michael De Vlieger_, Feb 07 2019: (Start)
11 (binary 1011) has more 1's than 0's, thus it is in the sequence.
12 (binary 1100) has an equal number of 0's and 1's, thus it is not in the sequence.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..14031 (all terms k <= 2^14).
Jason Bell, Thomas Finn Lidbetter, and Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A072600 (#0's < #1's), A072601 (#0's <= #1's), A031443 (#0's = #1's).

Cf. A072602 (#0's >= #1's), A072603 (#0's > #1's), this sequence (#0's <> #1's).

Select[Range@ 77, UnsameQ @@ DigitCount[#, 2] &] (* Michael De Vlieger, Feb 07 2019 *)

is(n)=2*hammingweight(n)!=exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

a(n) ~ n. - Charles R Greathouse IV, Apr 18 2020

A374664 Nonnegative numbers whose binary expansion has no ones in common with some of its cyclic shifts.

Original entry on oeis.org

0, 2, 4, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 35, 36, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 72, 73, 74, 76, 80, 82, 84, 96, 97, 100, 112, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 140, 144, 145, 146, 148, 150, 152, 153, 160, 161, 162

Offset: 1

Rémy Sigrist, Jul 15 2024

Leading zeros in binary expansions are ignored.
All positive terms belong to A072602.
A number k belongs to the sequence iff A001196(k) belongs to the sequence.

			The first terms, with their binary expansion and an appropriate cyclic shift, are:
  n   a(n)  bin(a(n))  cyc
  --  ----  ---------  ------
   1     0          0       0
   2     2         10      01
   3     4        100     001
   4     8       1000    0001
   5     9       1001    0110
   6    10       1010    0101
   7    12       1100    0011
   8    16      10000   00001
   9    17      10001   00110
  10    18      10010   00101
  11    20      10100   01001
  12    24      11000   00011
  13    32     100000  000001
  14    33     100001  000110
  15    34     100010  000101
  16    35     100011  011100

Index entries for sequences related to binary expansion of n

Cf. A001196, A006257, A038572, A072602, A140900, A374712.

is(n) = { my (x = max(exponent(n), 0), s = n); for (i = 0, x, s = (s >> 1) + if (s%2, 2^x, 0); if (bitand(s, n)==0, return (1););); return (0); }

A093327 Smallest sequence of numbers having in their binary representation alternately more (>) or not-more (<=) binary ones than zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 11, 12, 13, 16, 19, 20, 21, 24, 25, 32, 39, 40, 43, 44, 45, 48, 51, 52, 53, 56, 57, 64, 71, 72, 75, 76, 77, 80, 83, 84, 85, 88, 89, 96, 99, 100, 101, 104, 105, 112, 113, 128, 143, 144, 151, 152, 155, 156, 157, 160, 167, 168, 171, 172, 173, 176

Offset: 0

Reinhard Zumkeller, Jul 12 2004

nonn

A000120(a(n)) < A023416(a(n)) iff A000120(a(n+1))>=A023416(a(n+1));

A037861(a(2*n)) >= 0, A037861(a(2*n+1)) < 0.

Cf. A007088, A072602, A072602.

cp's OEIS Frontend

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

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A072600 Numbers which in base 2 have fewer 0's than 1's.

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A072603 Numbers which in base 2 have more 0's than 1's.

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A072601 Numbers which in base 2 have at least as many 1's as 0's.

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A044951 Numbers having a different number of ones and zeros in base 2.

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A374664 Nonnegative numbers whose binary expansion has no ones in common with some of its cyclic shifts.

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PARI

A093327 Smallest sequence of numbers having in their binary representation alternately more (>) or not-more (<=) binary ones than zeros.

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