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 A053645 Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal. 85
 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Triangle read by rows in which row n lists the first 2^n nonnegative integers (A001477), n >= 0. Right border gives A000225. Row sums give A006516. See example. - Omar E. Pol, Oct 17 2013 Without the initial zero also: zeroless numbers in base 3 (A032924: 1, 2, 11, 12, 21, ...), ternary digits decreased by 1 and read as binary. - M. F. Hasler, Jun 22 2020 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, preprint, Theoretical Computer Sci., 98 (1992), 163-197. J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197 (see Ex. 24). Index entries for sequences related to binary expansion of n FORMULA a(n) = n - 2^A000523(n). G.f.: 1/(1-x) * ((2x-1)/(1-x) + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003 a(n) = (A006257(n)-1)/2. - N. J. A. Sloane, May 16 2003 a(1) = 0, a(2n) = 2a(n), a(2n+1) = 2a(n) + 1. - N. J. A. Sloane, Sep 13 2003 a(n) = A062050(n) - 1. - N. J. A. Sloane, Jun 12 2004 a(A004760(n+1)) = n. - Reinhard Zumkeller, May 20 2009 a(n) = f(n-1,1) with f(n,m) = if n < m then n else f(n-m,2*m). - Reinhard Zumkeller, May 20 2009 Conjecture: a(n) = (1 - A036987(n-1))*(1 + a(n-1)) for n > 1 with a(1) = 0. - Mikhail Kurkov, Jul 16 2019 EXAMPLE From Omar E. Pol, Oct 17 2013: (Start) Written as an irregular triangle the sequence begins: 0; 0,1; 0,1,2,3; 0,1,2,3,4,5,6,7; 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15; ... (End) MAPLE seq(n - 2^ilog2(n), n=1..1000); # Robert Israel, Dec 23 2015 MATHEMATICA Table[n - 2^Floor[Log2[n]], {n, 100}] (* IWABUCHI Yu(u)ki, May 25 2017 *) Table[FromDigits[Rest[IntegerDigits[n, 2]], 2], {n, 100}] (* IWABUCHI Yu(u)ki, May 25 2017 *) PROG (Haskell) a053645 1 = 0 a053645 n = 2 * a053645 n' + b where (n', b) = divMod n 2 -- Reinhard Zumkeller, Aug 28 2014 a053645_list = concatMap (0 `enumFromTo`) a000225_list -- Reinhard Zumkeller, Feb 04 2013, Mar 23 2012 (PARI) a(n)=n-2^(#binary(n)-1) \\ Charles R Greathouse IV, Sep 02 2015 (Magma) [n - 2^Ilog2(n): n in [1..70]]; // Vincenzo Librandi, Jul 18 2019 (Python) def a(n): return n - 2**(n.bit_length()-1) print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2021 (Python) def A053645(n): return n&(1<

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Last modified August 8 02:48 EDT 2024. Contains 375018 sequences. (Running on oeis4.)