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 A070939 Length of binary representation of n. 468
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 OFFSET 0,3 COMMENTS Zero is assumed to be represented as 0. For n>1, n appears 2^(n-1) times. - Lekraj Beedassy, Apr 12 2006 a(n) is the permanent of the n X n 0-1 matrix whose (i,j) entry is 1 iff i=1 or i=j or i=2*j. For example, a(4)=3 is per([[1, 1, 1, 1], [1, 1, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]). - David Callan, Jun 07 2006 a(n) is the number of different contiguous palindromic bit patterns in the binary representation of n; for examples, for 5=101_2 the bit patterns are 0, 1, 101; for 7=111_2 the corresponding patterns are 1, 11, 111; for 13=1101_2 the patterns are 0, 1, 11, 101. - Hieronymus Fischer, Mar 13 2012 A103586(n) = a(n + a(n)); a(A214489(n)) = A103586(A214489(n)). - Reinhard Zumkeller, Jul 21 2012 Number of divisors of 2^n that are <= n. - Clark Kimberling, Apr 21 2019 REFERENCES G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255. L. Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127. LINKS T. D. Noe, Table of n, a(n) for n = 0..1024 K. Hessami Pilehrood, T. Hessami Pilehrood, Vacca-Type Series for Values of the Generalized Euler Constant Function and its Derivative, J. Integer Sequences, 13 (2010), #10.7.3. L. Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004. R. Stephan, Some divide-and-conquer sequences ... R. Stephan, Table of generating functions R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003. FORMULA a(0) = 1; for n >= 1, a(n) = 1 + floor(log_2(n)) = 1 + A000523(n). G.f.: 1 + 1/(1-x) * Sum(k>=0, x^2^k). - Ralf Stephan, Apr 12 2002 a(0)=1, a(1)=1 and a(n) = 1+a(floor(n/2)). - Benoit Cloitre, Dec 02 2003 a(n) = A000120(n) + A023416(n). - Lekraj Beedassy, Apr 12 2006 a(2^m + k) = m + 1, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 14 2017 a(n) = A113473(n) if n>0. EXAMPLE 8 = 1000 in binary has length 4. MAPLE A070939 := n -> `if`(n=0, 1, ilog2(2*n)): seq(A070939(n), n=0..104); # revised by Peter Luschny, Aug 10 2017 MATHEMATICA Table[Length[IntegerDigits[n, 2]], {n, 0, 50}] (* Stefan Steinerberger, Apr 01 2006 *) Join[{1}, IntegerLength[Range, 2]] (* Harvey P. Dale, Aug 18 2013 *) a[ n_] := If[ n < 1, Boole[n == 0], BitLength[n]]; (* Michael Somos, Jul 10 2018 *) PROG (MAGMA) A070939:=func< n | n eq 0 select 1 else #Intseq(n, 2) >; [ A070939(n): n in [0..104] ]; // Klaus Brockhaus, Jan 13 2011 (PARI) {a(n) = if( n<1, n==0, #binary(n))} /* Michael Somos, Aug 31 2012 */ (PARI) apply( {A070939(n)=exponent(n+!n)+1}, [0..99]) \\ works for negative n and is much faster than the above. - M. F. Hasler, Jan 04 2014, updated Feb 29 2020 (Haskell) a070939 n = if n < 2 then 1 else a070939 (n `div` 2) + 1 a070939_list = 1 : 1 : l  where    l bs = bs' ++ l bs' where bs' = map (+ 1) (bs ++ bs) -- Reinhard Zumkeller, Jul 19 2012, Jun 07 2011 (Sage) def A070939(n) : return (2*n).exact_log(2) if n != 0 else 1 [A070939(n) for n in range(100)] # Peter Luschny, Aug 08 2012 CROSSREFS Cf. A070940, A070941, A001511, A000523. A029837(n+1) gives the length of binary representation of n without the leading zeros (i.e., when zero is represented as the empty sequence). For n>0 this is equal to a(n). This is Guy Steele's sequence GS(4, 4) (see A135416). Cf. A000120, A007088, A023416, A059015, A113473. Cf. A083652 (partial sums). Sequence in context: A237261 A004258 A029837 * A113473 A265370 A238407 Adjacent sequences:  A070936 A070937 A070938 * A070940 A070941 A070942 KEYWORD nonn,easy,nice,core AUTHOR N. J. A. Sloane, May 18 2002 EXTENSIONS a(4) corrected by Antti Karttunen, Feb 28 2003 STATUS approved

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Last modified July 3 20:41 EDT 2020. Contains 335418 sequences. (Running on oeis4.)