A004233 -id:A004233 - cp's OEIS frontend

A000523 a(n) = floor(log_2(n)).

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

Offset: 1

N. J. A. Sloane

Or, n >= 0 appears 2^n times. - Jon Perry, Sep 21 2002
a(n) + 1 = number of bits in binary expansion of n.
Largest power of 2 dividing lcm(1..n): A007814(A003418(n)).
log_2(0) = -infinity.
Also Max_{k=1..n} Omega(k), where Omega(n) = A001222(n), number of prime factors with repetition; see A080613. - Reinhard Zumkeller, Feb 25 2003
From Paul Weisenhorn, Sep 29 2010, updated Aug 11 2020: (Start)
Arithmetic mean: m(1,(c+1)/c) = (2*c+1)/(2*c); harmonic mean: h(1,(c+1)/c) = 2*(c+1)/(2*c+1);
a(n) is the number of means to reach (n+1)/n from 2/1; with m for 0 and h for 1, the inverse binary expansion of n, without the leading 1, gives the sequence of means.
For example, n=20; inverse binary expansion without the leading 1: 0010 ---> m m h m or m(1, m(1, h(1, m(1, 2)))) = 21/20.
The 4 twofold means for n from 4 to 7:
m(1,m(1,2)) = m(1,3/2) = 5/4,
h(1,m(1,2)) = h(1,3/2) = 6/5,
m(1,h(1,2)) = m(1,4/3) = 7/6,
h(1,h(1,2)) = h(1,4/3) = 8/7. (End) [Edited by Petros Hadjicostas, Jul 23 2020]
As function of the absolute value, defines the minimal Euclidean function v on Z\{0}. A ring R is Euclidean if for some function v : R\{0}->N a division by nonzero b can be defined with remainder r satisfying either r=0 or v(r) < v(b). For the integers taking v(n)=|n| works, but v(n) = floor(log_2(|n|)) works as well; moreover it is the possibility with smallest possible values. For division by b>0 one can always choose |r| <= floor(b/2); this sequence satisfies a(1) = 0 and recursively a(n) = 1 + max(a(1), ..., a(floor(n/2))) for n > 1. - Marc A. A. van Leeuwen, Feb 16 2011
Maximum number of guesses required to find any k in a range of 1..n, with 'higher', 'lower' and 'correct' as answers. - Jon Perry, Nov 02 2013
Number of powers of 2 <= n. - Ralph-Joseph Tatt, Apr 23 2018
a(n) + 1 is the minimum number of pairwise disjoint subsets of an n-element set such that for each k from 1 to n there is a set with cardinality k which is the union of some of those subsets. - Wojciech Raszka, Apr 15 2019
Minimum height of an n-node binary tree. - Yuchun Ji, Mar 22 2021

			a(5)=2 because the binary expansion of 5 (=101) has three bits.

Rüdeger Baumann, Computer-Knobelei, LOG IN Heft 159 (2009), 74-77. - Paul Weisenhorn, Sep 29 2010
G. H. Hardy, Note on Dr. Vacca's series for gamma, Quart. J. Pure Appl. Math., Vol. 43 (1912), pp. 215-216.
Ernst Jacobsthal, Über die Eulersche konstante, Mathematisch-Naturwissenschaftliche Blätter, Vol. 3, No. 9 (1906), pp. 153-154.
Donald E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, p. 400.
Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.1.3, Problem 41, p. 589. - From N. J. A. Sloane, Aug 03 2012

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
G. H. Hardy, Note on Dr. Vacca's series for gamma, Quart. J. Pure Appl. Math. 43 (1912), 215-216. [Available only in the USA through the Hathi Trust.]
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).
G. Vacca, A new series for the Eulerian constant gamma=.577..., Quart. J. Pure Appl. Math., Vol. 41 (1910), pp. 363-368.

Cf. A000193, A000195, A001222, A001620, A003462, A004233, A029837, A032924, A061168 (partial sums), A070939, A081604, A107680, A113473, A152487, A240857.

a000523 1 = 0
a000523 n = 1 + a000523 (div n 2)
a000523_list = 0 : f [0] where
   f xs = ys ++ f ys where ys = map (+ 1) (xs ++ xs)
-- Reinhard Zumkeller, Dec 31 2012, Feb 04 2012, Mar 18 2011

Magma
```
[Ilog2(n) : n in [1..130] ];
```

A000523 := proc(n)
    ilog2(n) ;
end proc: # R. J. Mathar, Nov 28 2016
seq(A000523(n), n=1..90);

Floor[Log[2,Range[110]]] (* Harvey P. Dale, Jul 16 2012 *)
a[ n_] := If[ n < 1, 0, BitLength[n] - 1]; (* Michael Somos, Jul 10 2018 *)

{a(n) = floor(log(n) / log(2))} \\ Likely to yield incorrect results for many if not almost all n. Better use most recent code.

{a(n) = if( n<1, 0, #binary(n) - 1)}; /* Michael Somos, May 28 2014 */

a(n)=logint(n,2) \\ Charles R Greathouse IV, Sep 01 2015

a(n)=exponent(n) \\ Charles R Greathouse IV, Nov 09 2017

def A000523(n):
    return len(bin(n))-3 # Chai Wah Wu, Jul 09 2020

def a(n): return n.bit_length() - 1
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Apr 18 2023

a(n) = A070939(n) - 1 for n >= 1.
a(n) = if n > 1, then a(floor(n / 2)) + 1; else 0. - Reinhard Zumkeller, Oct 29 2001
G.f.: (1/(1 - x)) * Sum_{k>=1} x^2^k. - Ralf Stephan, Apr 13 2002
a(n+1) = number of digits of n-th number with no 0 in ternary representation = A081604(A032924(n)); A107680(n) = A003462(a(n+1)). - Reinhard Zumkeller, May 20 2005
a(n) = A152487(n-1,0) = A152487(n,1). - Reinhard Zumkeller, Dec 06 2008
a(n) = k with 2^k <= n < 2^(k+1); a(n) = floor(log_2(n)). - Paul Weisenhorn, Sep 29 2010
a(n) = Max_{k=1..n} A240857(n,k). - Reinhard Zumkeller, Apr 14 2014
a(n) = A113473(n) - 1. - Filip Zaludek, Oct 29 2016
Sum_{n>=2} (-1)^n*a(n)/n = gamma = A001620 (Jacobsthal, 1906; Vacca, 1910). - Amiram Eldar, Jun 12 2021
a(n) = floor(Sum_{k=1..n-1} (n+1)^(n-2^k)) mod n. - Joseph M. Shunia, Jul 19 2024

Error in 4th term, pointed out by Joe Keane (jgk(AT)jgk.org), has been corrected.

More terms from Michael Somos, Aug 02 2002

A029837 Binary order of n: log_2(n) rounded up to next integer.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

N. J. A. Sloane

Or, ceiling(log_2(n)).
Worst-case cost of binary search.
Equal to number of binary digits in n unless n is a power of 2 when it is one less.
Thus a(n) gives the length of the binary representation of n - 1 (n >= 2), which is also A070939(n - 1).
Let x(0) = n > 1 and x(k + 1) = x(k) - floor(x(k)/2), then a(n) is the smallest integer such that x(a(n)) = 1. - Benoit Cloitre, Aug 29 2002
Also number of division steps when going from n to 1 by process of adding 1 if odd, or dividing by 2 if even. - Cino Hilliard, Mar 25 2003
Number of ways to write n as (x + 2^y), x >= 0. Number of ways to write n + 1 as 2^x + 3^y (cf. A004050). - Benoit Cloitre, Mar 29 2003
The minimum number of cuts for dividing an object into n (possibly unequal) pieces. - Karl Ove Hufthammer (karl(AT)huftis.org), Mar 29 2010
Partial sums of A209229; number of powers of 2 not greater than n. - Reinhard Zumkeller, Mar 07 2012

			a(1) = 0, since log_2(1) = 0.
a(2) = 1, since log_2(2) = 1.
a(3) = 2, since log_2(3) = 1.58...
a(n) = 7 for n = 65, 66, ..., 127, 128.
G.f. = x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + ... - _Michael Somos_, Jun 02 2019

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, p. 70.
G. J. E. Rawlins, Compared to What? An Introduction to the Analysis of Algorithms, W. H. Freeman, 1992; see pp. 108, 118.

T. D. Noe, Table of n, a(n) for n = 1..10000
Cino Hilliard, The x+1 conjecture [broken link]
Lionel Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.
Eric Weisstein's World of Mathematics, Bit Length.
Index entries for sequences related to binary expansion of n

Cf. A000523, A070939, A000193, A000195, A004233, A113473, A053644.
Partial sums of A036987.
Used for several definitions: A029827, A036378-A036390. Partial sums: A001855.

a029837 n = a029837_list !! (n-1)
a029837_list = scanl1 (+) a209229_list
-- Reinhard Zumkeller, Mar 07 2012
(Common Lisp) (defun A029837 (n) (integer-length (1- n))) ; James Spahlinger, Oct 15 2012

[Ceiling(Log(2, n)): n in [1..100]]; // Vincenzo Librandi, Jun 14 2019

a:= n-> (p-> p+`if`(2^pAlois P. Heinz, Mar 18 2013

a[n_] := Ceiling[Log[2, n]]; Array[a, 105] (* Robert G. Wilson v, Dec 09 2005 *)
Table[IntegerLength[n - 1, 2], {n, 1, 105}] (* Peter Luschny, Dec 02 2017 *)
a[n_] := If[n < 1, 0, BitLength[n - 1]]; (* Michael Somos, Jul 10 2018 *)
Join[{0}, IntegerLength[Range[130], 2]] (* Vincenzo Librandi, Jun 14 2019 *)

{a(n) = if( n<1, 0, ceil(log(n) / log(2)))};

/* Set p = 1, then: */
xpcount(n,p) = for(x=1, n, p1 = x; ct=0; while(p1>1, if(p1%2==0,p1/=2; ct++,p1 = p1*p+1)); print1(ct, ", "))

{a(n) = if( n<2, 0, exponent(n-1)+1)}; /* Michael Somos, Jul 10 2018 */

def A029837(n):
    s = bin(n)[2:]
    return len(s) - (1 if s.count('1') == 1 else 0) # Chai Wah Wu, Jul 09 2020

def A029837(n): return (n-1).bit_length() # Chai Wah Wu, Jun 30 2022

(1 to 80).map(n => Math.ceil(Math.log(n)/Math.log(2)).toInt) // Alonso del Arte, Feb 19 2020

a(n) = ceiling(log_2(n)).
a(1) = 0; for n > 1, a(2n) = a(n) + 1, a(2n + 1) = a(n) + 1. Alternatively, a(1) = 0; for n > 1, a(n) = a(ceiling(n/2)) + 1. [corrected by Ilya Gutkovskiy, Mar 21 2020]
a(n) = k such that n^(1/k - 1) > 2 > n^(1/k), or the least value of k for which floor n^(1/k) = 1. a(n) = k for all n such that 2^(k - 1) < n < 2^k. - Amarnath Murthy, May 06 2001
G.f.: x/(1 - x) * Sum_{k >= 0} x^2^k. - Ralf Stephan, Apr 13 2002
A062383(n-1) = 2^a(n). - Johannes W. Meijer, Jul 06 2009
a(n+1) = -Sum_{k = 1..n} mu(2*k)*floor(n/k). - Benoit Cloitre, Oct 21 2009
a(n+1) = A113473(n). - Michael Somos, Jun 02 2019

Additional comments from Daniele Parisse

More terms from Michael Somos, Aug 02 2002

A096509 Number of prime-powers [including primes] in the (up and down) neighborhood of n with Ceiling[Log[n]] radius.

Original entry on oeis.org

0, 2, 4, 4, 4, 4, 4, 5, 4, 5, 4, 3, 3, 4, 3, 4, 3, 3, 3, 3, 4, 3, 4, 3, 4, 4, 5, 5, 5, 4, 4, 3, 4, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 2, 1, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 4, 3, 4, 4, 3, 3, 3, 2, 3, 3, 4, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4

Offset: 1

Labos Elemer, Jul 12 2004

nonn

With increasing n the radius of log(n) slowly increases, while frequency of prime-powers decreases. Thus hard to estimate upper bound of terms in this sequence.

Heuristically a(n) = 0 about 1/e^2 = 13.53...% of the time. The first few instances are 1, 300, 324, 895, 896, 897, 898, 899, 1077, .... - Charles R Greathouse IV, Apr 30 2015

			n=284736: in [284723,284749] around n, 8 prime(powers) occur,radius=13, a[284736]=8.

Cf. A096510, A096511, A096512.

a[n_] := Select[Range[n - Ceiling[Log[n]], n + Ceiling[Log[n]]], PrimePowerQ] // Length; Array[a, 105] (* Jean-François Alcover, Oct 06 2016 *)

a(n)=my(t=ceil(log(n))); sum(k=n-t,n+t,!!isprimepower(k)) \\ Charles R Greathouse IV, Apr 29 2015

a(n) <= A023193(2*A004233(n)+1) + A000720(A000523(A004233(n) + n)) and so a(n) << log n/log log n (with constant at most 4 + 1/log(2) = 5.442...). Probably a(n) < 2 log n/log log n + O(log n/(log log n)^2). - Charles R Greathouse IV, Apr 29 2015

A000195 a(n) = floor(log(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

N. J. A. Sloane

Equals A004233(n) - 1 for n > 1.

Does not satisfy Benford's law [Whyman et al., 2016] - N. J. A. Sloane, Feb 12 2017

T. D. Noe, Table of n, a(n) for n=1..10000
G. Whyman, N. Ohtori, E. Shulzinger, and Ed. Bormashenko, Revisiting the Benford law: When the Benford-like distribution of leading digits in sets of numerical data is expectable?, Physica A: Statistical Mechanics and its Applications, 461, 595-601 (2016).
Index entries for sequences related to Benford's law

Cf. A000193 (nearest integer to log(n)), A004233.

Cf. A000523.

a000195 = floor . log . fromIntegral  -- Reinhard Zumkeller, Mar 17 2015

Digits := 100; f := n->floor(evalf(log(n))); [ seq(f(n), n=1..100) ];

Floor@ Log@ Range@ 105 (* Michael De Vlieger, Aug 21 2017 *)

PARI
```
a(n)=floor(log(n))
```

Conjecture: a(n) = floor(3*n^2*(n^(1/(3*n^2))-1)), checked for n <= 10^6. - Joseph M. Shunia, Aug 02 2024

cp's OEIS Frontend

A000523 a(n) = floor(log_2(n)).

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A029837 Binary order of n: log_2(n) rounded up to next integer.

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A096509 Number of prime-powers [including primes] in the (up and down) neighborhood of n with Ceiling[Log[n]] radius.

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A000195 a(n) = floor(log(n)).

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A000193 Nearest integer to log n.

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