0,4

The binary weight of n is also called Hamming weight of n. [The term "Hamming weight" was named after the American mathematician Richard Wesley Hamming (1915-1998). - Amiram Eldar, Jun 16 2021]

a(n) is also the largest integer such that 2^a(n) divides binomial(2n, n) = A000984(n). - Benoit Cloitre, Mar 27 2002

To construct the sequence, start with 0 and use the rule: If k >= 0 and a(0), a(1), ..., a(2^k-1) are the first 2^k terms, then the next 2^k terms are a(0) + 1, a(1) + 1, ..., a(2^k-1) + 1. - Benoit Cloitre, Jan 30 2003

An example of a fractal sequence. That is, if you omit every other number in the sequence, you get the original sequence. And of course this can be repeated. So if you form the sequence a(0 * 2^n), a(1 * 2^n), a(2 * 2^n), a(3 * 2^n), ... (for any integer n > 0), you get the original sequence. - Christopher.Hills(AT)sepura.co.uk, May 14 2003

The n-th row of Pascal's triangle has 2^k distinct odd binomial coefficients where k = a(n) - 1. - Lekraj Beedassy, May 15 2003

Fixed point of the morphism 0 -> 01, 1 -> 12, 2 -> 23, 3 -> 34, 4 -> 45, etc., starting from a(0) = 0. - Robert G. Wilson v, Jan 24 2006

a(n) is the number of times n appears among the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner, Jan 25 2006

a(n) is the number of solutions of the Diophantine equation 2^m*k + 2^(m-1) + i = n, where m >= 1, k >= 0, 0 <= i < 2^(m-1); a(5) = 2 because only (m, k, i) = (1, 2, 0) [2^1*2 + 2^0 + 0 = 5] and (m, k, i) = (3, 0, 1) [2^3*0 + 2^2 + 1 = 5] are solutions. - Hieronymus Fischer, Jan 31 2006

The first appearance of k, k >= 0, is at a(2^k-1). - Robert G. Wilson v, Jul 27 2006

Sequence is given by T^(infinity)(0) where T is the operator transforming any word w = w(1)w(2)...w(m) into T(w) = w(1)(w(1)+1)w(2)(w(2)+1)...w(m)(w(m)+1). I.e., T(0) = 01, T(01) = 0112, T(0112) = 01121223. - Benoit Cloitre, Mar 04 2009

For n >= 2, the minimal k for which a(k(2^n-1)) is not multiple of n is 2^n + 3. - Vladimir Shevelev, Jun 05 2009

Triangle inequality: a(k+m) <= a(k) + a(m). Equality holds if and only if C(k+m, m) is odd. - Vladimir Shevelev, Jul 19 2009

The number of occurrences of value k in the first 2^n terms of the sequence is equal to binomial(n, k), and also equal to the sum of the first n - k + 1 terms of column k in the array A071919. Example with k = 2, n = 7: there are 21 = binomial(7,2) = 1 + 2 + 3 + 4 + 5 + 6 2's in a(0) to a(2^7-1). - Brent Spillner (spillner(AT)acm.org), Sep 01 2010, simplified by R. J. Mathar, Jan 13 2017

Let m be the number of parts in the listing of the compositions of n as lists of parts in lexicographic order, a(k) = n - length(composition(k)) for all k < 2^n and all n (see example); A007895 gives the equivalent for compositions into odd parts. - Joerg Arndt, Nov 09 2012

From Daniel Forgues, Mar 13 2015: (Start)

Just tally up row k (binary weight equal k) from 0 to 2^n - 1 to get the binomial coefficient C(n,k). (See A007318.)

0 1 3 7 15

0: O | . | . . | . . . . | . . . . . . . . |

1: | O | O . | O . . . | O . . . . . . . |

2: | | O | O O . | O O . O . . . |

3: | | | O | O O O . |

4: | | | | O |

Due to its fractal nature, the sequence is quite interesting to listen to.

(End)

The binary weight of n is a particular case of the digit sum (base b) of n. - Daniel Forgues, Mar 13 2015

The mean of the first n terms is 1 less than the mean of [a(n+1),...,a(2n)], which is also the mean of [a(n+2),...,a(2n+1)]. - Christian Perfect, Apr 02 2015

a(n) is also the largest part of the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2, 2, 2, 1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 20 2017

a(n) is also known as the population count of the binary representation of n. - Chai Wah Wu, May 19 2020

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 119.

Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.1.3, Problem 41, p. 589. - N. J. A. Sloane, Aug 03 2012

Manfred R. Schroeder, Fractals, Chaos, Power Laws. W.H. Freeman, 1991, p. 383.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Franklin T. Adams-Watters, and Frank Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS, Vol. 12 (2009), Article 09.5.6.

J.-P. Allouche, On an Inequality in a 1970 Paper of R. L. Graham, INTEGERS 21A (2021), #A2.

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98 (1992), pp. 163-197. (PS file on author's web page.)

Jean-Paul Allouche, Jeffrey Shallit and Jonathan Sondow, Summation of Series Defined by Counting Blocks of Digits, arXiv:math/0512399 [math.NT], 2005-2006.

Jean-Paul Allouche, Jeffrey Shallit and Jonathan Sondow, Summation of series defined by counting blocks of digits, J. Number Theory, Vol. 123 (2007), pp. 133-143.

Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., Vol. 49, No. 2 (1948), pp. 333-340. - N. J. A. Sloane, Mar 12 2009

Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.

Karl Dilcher and Larry Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, Vol. 22 (2015), #P2.24.

Josef Eschgfäller and Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.

Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.

Steven R. Finch, Pascal Sebah and Zai-Qiao Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Philippe Flajolet, Peter Grabner, Peter Kirschenhofer, Helmut Prodinger and Robert F. Tichy, Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci., Vol. 123, No. 2 (1994), pp. 291-314.

Michael Gilleland, Some Self-Similar Integer Sequences.

P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.

Ronald L. Graham, On primitive graphs and optimal vertex assignments, Internat. Conf. Combin. Math. (New York, 1970), Annals of the NY Academy of Sciences, Vol. 175, 1970, pp. 170-186.

Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, Vacca-Type Series for Values of the Generalized Euler Constant Function and its Derivative, J. Integer Sequences, Vol. 13 (2010), Article 10.7.3.

Nick Hobson, Python program for this sequence.

Kathy Q. Ji and Herbert S. Wilf, Extreme Palindromes, arXiv:math/0611465 [math.CO], 2006.

Guy Louchard and Helmut Prodinger, The Largest Missing Value in a Composition of an Integer and Some Allouche-Shallit-Type Identities, J. Int. Seq., Vol. 16 (2013), Article 13.2.2.

J.-L. Mauclaire and Leo Murata, On q-additive functions, I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.

J.-L. Mauclaire and Leo Murata, On q-additive functions, II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.

M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., Vol. 3 (1974), pp. 255-261.

Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.

Sam Northshield, Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,..., Amer. Math. Month., Vol. 117, No. 7 (2010), pp. 581-598.

Theophanes E. Raptis, Finite Information Numbers through the Inductive Combinatorial Hierarchy, arXiv:1805.06301 [physics.gen-ph], 2018.

Carlo Sanna, On Arithmetic Progressions of Integers with a Distinct Sum of Digits, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.1. - N. J. A. Sloane, Dec 29 2012

Nanci Smith, Problem B-82, Fib. Quart., Vol. 4, No. 4 (1966), pp. 374-375.

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, D. and G. Chudnovsky, eds., Springer, 2010, pp. 331-340.

Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.

Ralf Stephan, Table of generating functions.

Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Kenneth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., Vol. 32 (1977), pp. 717-730. See B(n). - N. J. A. Sloane, Apr 05 2014

J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Robert Walker, Self Similar Sloth Canon Number Sequences.

Eric Weisstein's World of Mathematics, Binary, Digit Count, Stolarsky-Harborth Constant, Digit Sum.

Wikipedia, Hamming weight.

Wolfram Research, Numbers in Pascal's triangle.

Index entries for "core" sequences

Index entries for sequences related to binary expansion of n

Index entries for Colombian or self numbers and related sequences

Index entries for sequences that are fixed points of mappings

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

a(0) = 0, a(2^i) = 1; otherwise if n = 2^i + j with 0 < j < 2^i, a(n) = a(j) + 1.

G.f.: Product_{k >= 0} (1 + y*x^(2^k)) = Sum_{n >= 0} y^a(n)*x^n. - N. J. A. Sloane, Jun 04 2009

a(n) = a(n-1) + 1 - A007814(n) = log_2(A001316(n)) = 2n - A005187(n) = A070939(n) - A023416(n). - Henry Bottomley, Apr 04 2001; corrected by Ralf Stephan, Apr 15 2002

a(n) = log_2(A000984(n)/A001790(n)). - Benoit Cloitre, Oct 02 2002

For n > 0, a(n) = n - Sum_{k=1..n} A007814(k). - Benoit Cloitre, Oct 19 2002

a(n) = n - Sum_{k>=1} floor(n/2^k) = n - A011371(n). - Benoit Cloitre, Dec 19 2002

G.f.: (1/(1-x)) * Sum_{k>=0} x^(2^k)/(1+x^(2^k)). - Ralf Stephan, Apr 19 2003

a(0) = 0, a(n) = a(n - 2^floor(log_2(n))) + 1. Examples: a(6) = a(6 - 2^2) + 1 = a(2) + 1 = a(2 - 2^1) + 1 + 1 = a(0) + 2 = 2; a(101) = a(101 - 2^6) + 1 = a(37) + 1 = a(37 - 2^5) + 2 = a(5 - 2^2) + 3 = a(1 - 2^0) + 4 = a(0) + 4 = 4; a(6275) = a(6275 - 2^12) + 1 = a(2179 - 2^11) + 2 = a(131 - 2^7) + 3 = a(3 - 2^1) + 4 = a(1 - 2^0) + 5 = 5; a(4129) = a(4129 - 2^12) + 1 = a(33 - 2^5) + 2 = a(1 - 2^0) + 3 = 3. - Hieronymus Fischer, Jan 22 2006

A fixed point of the mapping 0 -> 01, 1 -> 12, 2 -> 23, 3 -> 34, 4 -> 45, ... With f(i) = floor(n/2^i), a(n) is the number of odd numbers in the sequence f(0), f(1), f(2), f(3), f(4), f(5), ... - Philippe Deléham, Jan 04 2004

When read mod 2 gives the Morse-Thue sequence A010060.

Let floor_pow4(n) denote n rounded down to the next power of four, floor_pow4(n) = 4 ^ floor(log4 n). Then a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2, a(n) = a(floor(n / floor_pow4(n))) + a(n % floor_pow4(n)). - Stephen K. Touset (stephen(AT)touset.org), Apr 04 2007

a(n) = n - Sum_{k=2..n} Sum_{j|n, j >= 2} (floor(log_2(j)) - floor(log_2(j-1))). - Hieronymus Fischer, Jun 18 2007

a(n) = A138530(n, 2) for n > 1. - Reinhard Zumkeller, Mar 26 2008

a(A077436(n)) = A159918(A077436(n)); a(A000290(n)) = A159918(n). - Reinhard Zumkeller, Apr 25 2009

a(n) = A063787(n) - A007814(n). - Gary W. Adamson, Jun 04 2009

a(n) = A007814(C(2n, n)) = 1 + A007814(C(2n-1, n)). - Vladimir Shevelev, Jul 20 2009

For odd m >= 1, a((4^m-1)/3) = a((2^m+1)/3) + (m-1)/2 (mod 2). - Vladimir Shevelev, Sep 03 2010

a(n) - a(n-1) = { 1 - a(n-1) if and only if A007814(n) = a(n-1), 1 if and only if A007814(n) = 0, -1 for all other A007814(n) }. - Brent Spillner (spillner(AT)acm.org), Sep 01 2010

a(A001317(n)) = 2^a(n). - Vladimir Shevelev, Oct 25 2010

a(n) = A139351(n) + A139352(n) = Sum_k {A030308(n, k)}. - Philippe Deléham, Oct 14 2011

From Hieronymus Fischer, Jun 10 2012: (Start)

a(n) = Sum_{j = 1..m+1} (floor(n/2^j + 1/2)) - floor(n/2^j)), where m = floor(log_2(n)).

General formulas for the number of digits >= d in the base p representation of n, where 1 <= d < p: a(n) = Sum_{j = 1..m+1} (floor(n/p^j + (p-d)/p) - floor(n/p^j)), where m=floor(log_p(n)); g.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(d*p^j) - x^(p*p^j))/(1-x^(p*p^j)). (End)

a(n) = A213629(n, 1) for n > 0. - Reinhard Zumkeller, Jul 04 2012

a(n) = A240857(n,n). - Reinhard Zumkeller, Apr 14 2014

a(n) = log_2(C(2*n,n) - (C(2*n,n) AND C(2*n,n)-1)). - Gary Detlefs, Jul 10 2014

Sum_{n >= 1} a(n)/2n(2n+1) = (gamma + log(4/Pi))/2 = A344716, where gamma is Euler's constant A001620; see Sondow 2005, 2010 and Allouche, Shallit, Sondow 2007. - Jonathan Sondow, Mar 21 2015

For any integer base b >= 2, the sum of digits s_b(n) of expansion base b of n is the solution of this recurrence relation: s_b(n) = 0 if n = 0 and s_b(n) = s_b(floor(n/b)) + (n mod b). Thus, a(n) satisfies: a(n) = 0 if n = 0 and a(n) = a(floor(n/2)) + (n mod 2). This easily yields a(n) = Sum_{i = 0..floor(log_2(n))} (floor(n/2^i) mod 2). From that one can compute a(n) = n - Sum_{i = 1..floor(log_2(n))} floor(n/2^i). - Marek A. Suchenek, Mar 31 2016

Sum_{k>=1} a(k)/2^k = 2 * Sum_{k >= 0} 1/(2^(2^k)+1) = 2 * A051158. - Amiram Eldar, May 15 2020

Sum_{k>=1} a(k)/(k*(k+1)) = A016627 = log(4). - Bernard Schott, Sep 16 2020

a(m*(2^n-1)) >= n. Equality holds when 2^n-1 >= A000265(m), but also in some other cases, e.g., a(11*(2^2-1)) = 2 and a(19*(2^3-1)) = 3. - Pontus von Brömssen, Dec 13 2020

Using the formula a(n) = a(floor(n / floor_pow4(n))) + a(n mod floor_pow4(n)):

a(4) = a(1) + a(0) = 1,

a(8) = a(2) + a(0) = 1,

a(13) = a(3) + a(1) = 2 + 1 = 3,

a(23) = a(1) + a(7) = 1 + a(1) + a(3) = 1 + 1 + 2 = 4.

Gary W. Adamson points out (Jun 03 2009) that this can be written as a triangle:

0,

1,

1,2,

1,2,2,3,

1,2,2,3,2,3,3,4,

1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,

1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,

1,2,2,3,2,3,...

where the rows converge to A063787.

From Joerg Arndt, Nov 09 2012: (Start)

Connection to the compositions of n as lists of parts (see comment):

[ #]: a(n) composition

[ 0]: [0] 1 1 1 1 1

[ 1]: [1] 1 1 1 2

[ 2]: [1] 1 1 2 1

[ 3]: [2] 1 1 3

[ 4]: [1] 1 2 1 1

[ 5]: [2] 1 2 2

[ 6]: [2] 1 3 1

[ 7]: [3] 1 4

[ 8]: [1] 2 1 1 1

[ 9]: [2] 2 1 2

[10]: [2] 2 2 1

[11]: [3] 2 3

[12]: [2] 3 1 1

[13]: [3] 3 2

[14]: [3] 4 1

[15]: [4] 5

(End)

A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;

A000120 := n -> add(i, i=convert(n, base, 2)): # Peter Luschny, Feb 03 2011

with(Bits): p:=n->ilog2(n-And(n, n-1)): seq(p(binomial(2*n, n)), n=0..200) # Gary Detlefs, Jan 27 2019

Table[DigitCount[n, 2, 1], {n, 0, 105}]

Nest[Flatten[# /. # -> {#, # + 1}] &, {0}, 7] (* Robert G. Wilson v, Sep 27 2011 *)

Table[Plus @@ IntegerDigits[n, 2], {n, 0, 104}]

Nest[Join[#, # + 1] &, {0}, 7] (* IWABUCHI Yu(u)ki, Jul 19 2012 *)

Log[2, Nest[Join[#, 2#] &, {1}, 14]] (* gives 2^14 term, Carlos Alves, Mar 30 2014 *)

(PARI) {a(n) = if( n<0, 0, 2*n - valuation((2*n)!, 2))};

(PARI) {a(n) = if( n<0, 0, subst(Pol(binary(n)), x , 1))};

(PARI) {a(n) = if( n<1, 0, a(n\2) + n%2)}; /* Michael Somos, Mar 06 2004 */

(PARI) a(n)=my(v=binary(n)); sum(i=1, #v, v[i]) \\ Charles R Greathouse IV, Jun 24 2011

(PARI) a(n)=norml2(binary(n)) \\ better use {A000120=hammingweight}. - M. F. Hasler, Oct 09 2012, edited Feb 27 2020

(PARI) a(n)=hammingweight(n) \\ Michel Marcus, Oct 19 2013

(Common LISP) (defun floor-to-power (n pow) (declare (fixnum pow)) (expt pow (floor (log n pow)))) (defun enabled-bits (n) (if (< n 4) (n-th n (list 0 1 1 2)) (+ (enabled-bits (floor (/ n (floor-to-power n 4)))) (enabled-bits (mod n (floor-to-power n 4)))))) ; Stephen K. Touset (stephen(AT)touset.org), Apr 04 2007

See link in A139351 for Fortran program.

(Haskell)

import Data.Bits (Bits, popCount)

a000120 :: (Integral t, Bits t) => t -> Int

a000120 = popCount

a000120_list = 0 : c [1] where c (x:xs) = x : c (xs ++ [x, x+1])

-- Reinhard Zumkeller, Aug 26 2013, Feb 19 2012, Jun 16 2011, Mar 07 2011

(Haskell)

a000120 = concat r

where r = [0] : (map.map) (+1) (scanl1 (++) r)

-- Luke Palmer, Feb 16 2014

(Sage)

def A000120(n):

if n <= 1: return Integer(n)

return A000120(n//2) + n%2

[A000120(n) for n in range(105)] # Peter Luschny, Nov 19 2012

(Sage) def A000120(n) : return sum(n.digits(2)) # Eric M. Schmidt, Apr 26 2013

(Python) def A000120(n): return bin(n).count('1') # Chai Wah Wu, Sep 03 2014

(Python)

import numpy as np

A000120 = np.array([0], dtype="uint8")

for bitrange in range(25): A000120 = np.append(A000120, np.add(A000120, 1))

print([A000120[n] for n in range(0, 105)]) # Karl-Heinz Hofmann, Nov 07 2022

(Python) def A000120(n): return n.bit_count() # Requires Python 3.10 or higher. - Pontus von Brömssen, Nov 08 2022

(Python) Also see links.

(Scala) (0 to 127).map(Integer.bitCount(_)) // Alonso del Arte, Mar 05 2019

(Magma) [Multiplicity(Intseq(n, 2), 1): n in [0..104]]; // Marius A. Burtea, Jan 22 2020

(Magma) [&+Intseq(n, 2):n in [0..104]]; // Marius A. Burtea, Jan 22 2020

The basic sequences concerning the binary expansion of n are this one, A000788, A000069, A001969, A023416, A059015, A007088.

Partial sums see A000788. For run lengths see A131534. See also A001792, A010062.

Number of 0's in n: A023416 and A080791.

a(n) = n - A011371(n).

Sum of digits of n written in bases 2-16: this sequence, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.

Cf. A001620, A344716, A007814, A063787.

This is Guy Steele's sequence GS(3, 4) (see A135416).

Cf. A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564 (for base 10).

Cf. A230952 (boustrophedon transform).

Cf. A070939 (length of binary representation of n).

Sequence in context: A105056 A105061 A105164 * A105062 A106487 A105102

Adjacent sequences: A000117 A000118 A000119 * A000121 A000122 A000123

nonn,easy,core,nice,hear,look,base

N. J. A. Sloane

approved