A059014 -id:A059014 - cp's OEIS frontend

A059009 Numbers having an odd number of zeros in their binary expansion.

0, 2, 5, 6, 8, 11, 13, 14, 17, 18, 20, 23, 24, 27, 29, 30, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 128, 131

Offset: 0

Patrick De Geest, Dec 15 2000

Positions of ones in A059448 for n >= 1. - John Keith, Mar 09 2022

			18 is in the sequence because 18 = 10010_2. '10010' has three zeros. - _Indranil Ghosh_, Feb 04 2017

Indranil Ghosh, Table of n, a(n) for n = 0..25000 (terms 0..1000 from T. D. Noe)
Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.

Cf. A000069, A001969, A059010, A059011, A059012, A059013, A059014, A059448.

Cf. A023416.

a059009 n = a059009_list !! (n-1)
a059009_list = filter (odd . a023416) [1..]
-- Reinhard Zumkeller, Jan 21 2014

a:= proc(n) option remember;
  if n::even then -a(n/2) + 3*n + 1 else a((n-1)/2) + n + 1 fi
end proc:
a(0):= 0:
seq(a(n),n=0..100); # Robert Israel, Feb 23 2016

Select[Range[0,150],OddQ[Count[IntegerDigits[#,2],0]]&] (* Harvey P. Dale, Oct 22 2011 *)

is(n)=hammingweight(bitneg(n,#binary(n)))%2 \\ Charles R Greathouse IV, Mar 26 2013

a(n) = if(n==0,0, 2*n + (logint(n,2) - hammingweight(n) + 1) % 2); \\ Kevin Ryde, Mar 11 2021

i=j=0
while j<=800:
    if bin(i)[2:].count("0")%2:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 04 2017

maxrow <- 4 # by choice
onezeros <- 1
for(m in 1:(maxrow+1)){
  row <- onezeros[2^(m-1):(2^m-1)]
  onezeros <- c(onezeros, c(1-row, row) )
}
a <- which(onezeros == 0)
a
# Yosu Yurramendi, Mar 28 2017

a(0) = 0, a(2*n) = -a(n) + 6*n + 1, a(2*n+1) = a(n) + 2*n + 2. a(n) = 2*n + 1/2(1-(-1)^A023416(n)) = 2*n + A059448(n). - Ralf Stephan, Sep 17 2003

A059010 Natural numbers having an even number of nonleading zeros in their binary expansion.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 15, 16, 19, 21, 22, 25, 26, 28, 31, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 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, 129, 130

Offset: 0

Patrick De Geest, Dec 15 2000

Positions of ones in A298952, and of zeros in A059448. - John Keith, Mar 09 2022

Indranil Ghosh, Table of n, a(n) for n = 0..25000 (terms 0..1000 from T. D. Noe)
Jean Paul Allouche, Jeffrey Shallit, and Guentcho Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160. [From _N. J. A. Sloane_, Jan 31 2012]
Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.
Wadim Zudilin, A strange identity of an MF (Mahler function), arXiv:2403.13604 [math.NT], 2024.
Index entries for sequences related to binary expansion of n

Cf. A059009 (complement).
Cf. A023416 (number of 0-bits), A059448 (their parity), A298952 (opposite parity).
Cf.  A001969 (even 1-bits), A059012 (even both 0's and 1's), A059014 (even 0's, odd 1's).

a059010 n = a059010_list !! (n-1)
a059010_list = filter (even . a023416) [1..]
-- Reinhard Zumkeller, Jan 21 2014

Select[Range[130], EvenQ @ DigitCount[#, 2, 0] &] (* Jean-François Alcover, Apr 11 2011 *)

is(n)=hammingweight(bitneg(n,#binary(n)))%2==0 \\ Charles R Greathouse IV, Mar 26 2013

a(n) = if(n==0,1, 2*n + (logint(n,2) - hammingweight(n)) % 2); \\ Kevin Ryde, Mar 11 2021

#Program to generate the b-file
i=1
j=0
while j<=250:
    if bin(i)[2:].count("0")%2==0:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

maxrow <- 4 # by choice
onezeros <- 1
for(m in 1:(maxrow+1)){
  row <- onezeros[2^(m-1):(2^m-1)]
  onezeros <- c(onezeros, c(1-row, row) )
}
a <- which(onezeros == 1)
a
# Yosu Yurramendi, Mar 28 2017

a(0) = 1, a(2n) = -a(n) + 6n + 1, a(2n+1) = a(n) + 2n + 2. a(n) = 2n+1 - 1/2(1-(-1)^A023416(n)) = 2n+1 - A059448(n). - Ralf Stephan, Sep 17 2003

Name clarified by Antti Karttunen, Mar 28 2017

A059012 Numbers that have an even number of 0's and 1's in their binary expansion.

Original entry on oeis.org

3, 9, 10, 12, 15, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 129, 130, 132, 135, 136, 139, 141, 142, 144, 147, 149, 150, 153, 154, 156, 159, 160, 163, 165, 166, 169, 170, 172, 175, 177, 178, 180, 183, 184, 187, 189, 190, 192, 195, 197, 198

Offset: 1

Patrick De Geest, Dec 15 2000

Intersection of A001969 and A059010.

			36 is in the sequence because 36 = 100100_10. '100100' has two 1's and four 0's. - _Indranil Ghosh_, Feb 10 2017

Indranil Ghosh, Table of n, a(n) for n = 1..32768

Cf. A000069, A001969, A059009-A059014.

Select[Range[200],AllTrue[{DigitCount[#,2,0],DigitCount[#,2,1]},EvenQ]&] (* Harvey P. Dale, Aug 16 2021 *)

is(n)=hammingweight(n)%2==0 && hammingweight(bitneg(n, #binary(n)))%2==0 \\ Charles R Greathouse IV, Mar 26 2013

i=0
j=1
while j<=300:
    if bin(i)[2:].count("1")%2 == 0 == bin(i)[2:].count("0")%2:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 10 2017

A059011 Odd number of 0's and 1's in binary expansion.

Original entry on oeis.org

2, 8, 11, 13, 14, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 128, 131, 133, 134, 137, 138, 140, 143, 145, 146, 148, 151, 152, 155, 157, 158, 161, 162, 164, 167, 168, 171, 173, 174, 176, 179, 181, 182, 185, 186, 188, 191, 193, 194, 196, 199

Offset: 1

Patrick De Geest, Dec 15 2000

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A059009-A059014.

Cf. A000120, A023416, A071295.

a059011 n = a059011_list !! (n-1)
a059011_list = filter (odd . a071295) [0..]
-- Reinhard Zumkeller, Aug 09 2014

Select[Range[200],EvenQ[IntegerLength[#,2]]&&OddQ[DigitCount[#,2,1]]&] (* Harvey P. Dale, Oct 16 2012 *)

is(n)=hammingweight(n)%2 && hammingweight(bitneg(n, #binary(n)))%2 \\ Charles R Greathouse IV, Mar 26 2013

A071295(a(n)) is odd. - Reinhard Zumkeller, Aug 09 2014

A059013 Odd number of 0's and even number of 1's in binary expansion.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 23, 24, 27, 29, 30, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 257, 258, 260, 263, 264, 267, 269, 270, 272, 275, 277, 278, 281, 282, 284, 287

Offset: 1

Patrick De Geest, Dec 15 2000

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A059009-A059014.

Select[Range[0,300], EvenQ[DigitCount[#,2,1]] && OddQ[DigitCount[#,2,0]]&] (* Harvey P. Dale, Jun 13 2013 *)

is(n)=hammingweight(n)%2==0 && hammingweight(bitneg(n, #binary(n)))%2 \\ Charles R Greathouse IV, Mar 26 2013

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A059009 Numbers having an odd number of zeros in their binary expansion.

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A059010 Natural numbers having an even number of nonleading zeros in their binary expansion.

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A059012 Numbers that have an even number of 0's and 1's in their binary expansion.

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A059011 Odd number of 0's and 1's in binary expansion.

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A059013 Odd number of 0's and even number of 1's in binary expansion.

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Author

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Programs

Mathematica

PARI