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 A080791 Number of nonleading 0's in binary expansion of n. 51
 0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 1, 0, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS In this version we consider the number zero to have no nonleading 0's, thus a(0) = 0. The variant A023416 has a(0) = 1. Number of steps required to reach 1, starting at n + 1, under the operation: if x is even divide by 2 else add 1. This is the x + 1 problem (as opposed to the 3x + 1 problem). LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..10000 FORMULA From Antti Karttunen, Dec 12 2013: (Start) a(n) = A029837(n+1) - A000120(n). a(0) = 0, and for n > 0, a(n) = (a(n-1) + A007814(n) + A036987(n-1)) - 1. For all n >= 1, a(A054429(n)) = A048881(n-1) = A000120(n) - 1. Equally, for all n >= 1, a(n) = A000120(A054429(n)) - 1. (End) Recurrence: a(2n) = a(n) + 1 (for n > 0), a(2n + 1) = a(n). - Ralf Stephan from Cino Hillard's PARI program, Dec 16 2013. Corrected by Alonso del Arte, May 21 2017 after consultation with Chai Wah Wu and Ray Chandler, "n > 0" added by M. F. Hasler, Oct 26 2017 a(n) = A023416(n) for all n > 0. - M. F. Hasler, Oct 26 2017 G.f. g(x) satisfies g(x) = (1+x)*g(x^2) + x^2/(1-x^2). - Robert Israel, Oct 26 2017 EXAMPLE a(4) = 2 since 4 in binary is 100, which has two zeros. a(5) = 1 since 5 in binary is 101, which has only one zero. MAPLE 0, seq(numboccur(0, convert(n, base, 2)), n=1..100); # Robert Israel, Oct 26 2017 MATHEMATICA {0}~Join~Table[Last@ DigitCount[n, 2], {n, 120}] (* Michael De Vlieger, Mar 07 2016 *) f[n_] := If[OddQ@ n, f[n -1] -1, f[n/2] +1]; f[0] = f[1] = 0; Array[f, 105, 0] (* Robert G. Wilson v, May 21 2017 *) Join[{0}, Table[Count[IntegerDigits[n, 2], 0], {n, 1, 100}]] (* Vincenzo Librandi, Oct 27 2017 *) PROG (PARI) a(n)=if(n, a(n\2)+1-n%2) (PARI) A080791(n)=if(n, logint(n, 2)+1-hammingweight(n)) \\ M. F. Hasler, Oct 26 2017 (Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library) (define (A080791 n) (- (A029837 (+ 1 n)) (A000120 n))) ;; Alternative version based on a simple recurrence: (definec (A080791 n) (if (zero? n) 0 (+ (A080791 (- n 1)) (A007814 n) (A036987 (- n 1)) -1))) ;; from Antti Karttunen, Dec 12 2013 (Python) def a(n): return bin(n)[2:].count("0") if n>0 else 0 # Indranil Ghosh, Apr 10 2017 CROSSREFS Cf. A000120, A007814, A023416, A029837, A036987, A080791-A080801, A000120, A048881, A054429, A092339, A102364, A120511, A233271, A233272, A233273. Sequence in context: A050606 A277721 A023416 * A124748 A161225 A174980 Adjacent sequences:  A080788 A080789 A080790 * A080792 A080793 A080794 KEYWORD easy,nonn AUTHOR Cino Hilliard, Mar 25 2003 STATUS approved

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Last modified August 15 07:26 EDT 2018. Contains 313756 sequences. (Running on oeis4.)