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 A029837 Binary order of n: log_2(n) rounded up to next integer. 128
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 REFERENCES 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. LINKS 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. FORMULA 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 EXAMPLE 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... If a(n)=7, then 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 MAPLE a:= n-> (p-> p+`if`(2^p1, if(p1%2==0, p1/=2; ct++, p1 = p1*p+1)); print1(ct, ", ")) (PARI) {a(n) = if( n<2, 0, exponent(n-1)+1)}; /* Michael Somos, Jul 10 2018 */ (Haskell) 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 (MAGMA) [Ceiling(Log(2, n)): n in [1..100]]; // Vincenzo Librandi, Jun 14 2019 (Scala) (1 to 80).map(n => Math.ceil(Math.log(n)/Math.log(2)).toInt) // Alonso del Arte, Feb 19 2020 (Python) def A029837(n):     s = bin(n)[2:]     return len(s) - (1 if s.count('1') == 1 else 0) # Chai Wah Wu, Jul 09 2020 CROSSREFS Cf. A000523, A070939, A000193, A000195, A004233, A113473, A053644. Used for several definitions: A029827, A036378-A036390. Partial sums: A001855. Sequence in context: A075172 A237261 A004258 * A070939 A113473 A265370 Adjacent sequences:  A029834 A029835 A029836 * A029838 A029839 A029840 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Additional comments from Daniele Parisse (daniele.parisse(AT)m.dasa.de) More terms from Michael Somos, Aug 02 2002 STATUS approved

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Last modified September 20 23:24 EDT 2020. Contains 337265 sequences. (Running on oeis4.)