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 A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n. 227
 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS This sequence is an exception to my usual rule that when every other term of a sequence is 0 then those 0's should be omitted. In this case we would get A001511. - N. J. A. Sloane. To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre, Mar 06 2003 Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), ..., starting from a(1) = 0. - Philippe Deléham, Mar 15 2004 a(n) is also the number of times to repeat a step on an even number in the hailstone sequence referenced in the Collatz conjecture. - Alex T. Flood (whiteangelsgrace(AT)gmail.com), Sep 22 2006 Let F(n) be the n-th Fermat number (A000215). Then F(a(r-1)) divides F(n)+2^k for r = mod(k,2^n) and r != 1. - T. D. Noe, Jul 12 2007 The following relation holds: 2^A007814(n)*(2*A025480(n-1)+1) = A001477(n) = n. (See functions hd,tl and cons in [Paul Tarau 2009]). a(n) is the number of 0's at the end of n when n is written in base 2. a(n+1) is the number of 1's at the end of n when n is written in base 2. - M. F. Hasler, Aug 25 2012 Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 0).  That is, A003188(n) XOR A003188(n+1) == 2^A007814(n).  - Russ Cox, Dec 04 2010 The sequence is squarefree [Allouche and Shallit] a(n) is the number of zero coefficients in the n-th Stern polynomial, A125184. - T. D. Noe, Mar 01 2011 Lemma: For nr. Proof: We have n=b2^r and m=c2^r with br and n=1, a(A004760(n+1)) = a(n). - Vladimir Shevelev, Apr 15 2009 2^(a(n)) = A006519(n). - Philippe DELEHAM, Apr 22 2009 a(n) = A063787(n) - A000120(n) - Gary W. Adamson, Jun 04 2009 a(C(n,k)) = A000120(k)+A000120(n-k)-A000120(n). - Vladimir Shevelev, Jul 19 2009 a(n!) = n-A000120(n). - Vladimir Shevelev, Jul 20 2009 v_{2}(n) = sum_{r>=1} \frac{r}{2^{r+1}} sum_{k=0..2^{r+1}-1} e^{\frac{2(k*pi i(n+2^r)}{2^{r+1}}} - A. Neves, Sep 28 2010, corrected Oct 04 2010 a(n)(mod 2) = A096268(n-1). - Robert G. Wilson v, Jan 18 2012. a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7; a(n) = (A037227(n)-1)/2. - Reinhard Zumkeller, Jun 30 2012 a((2*n-1)*2^p) = p, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 04 2013 EXAMPLE 2^3 divides 24, so a(24)=3. Contribution from Omar E. Pol, Jun 12 2009: (Start) Triangle begins: 0; 1,0; 2,0,1,0; 3,0,1,0,2,0,1,0; 4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0; 5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0; 6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,... (End) MAPLE ord := proc(n) local i, j; if n=0 then RETURN(0); fi; i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end: seq(ord(n), n=1..111); A007814 := n -> padic[ordp](n, 2): seq(A007814(n), n=1..111); # [Peter Luschny, Nov 26 2010] MATHEMATICA Table[IntegerExponent[n, 2], {n, 64}] (Eric W. Weisstein) p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ] DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on the same concept: Floor[Log[2, BitXor[x, x - 1]]] (Jaume Simon Gispert (jaume(AT)nuem.com), Aug 29 2004) Nest[Join[ #, ReplacePart[ #, Length[ # ] -> Last[ # ] + 1]] &, {0, 1}, 5] (N. J. Gunther, May 23 2009) Nest[ Flatten[# /. a_Integer -> {0, a + 1}] &, {0}, 7] PROG (PARI) A007814(n)=valuation(n, 2); (Haskell) a007814 n = length $takeWhile ((== 0) . (mod n))$ iterate (* 2) 2 -- Reinhard Zumkeller, May 14 2011, Apr 08 2011 (Haskell) a007814 n | odd n = 0 | otherwise = 1 + a007814 (n div 2) (Matlab) %Input: %  n: an integer %Output: %  m: max power of 2 such that 2^m divides n %  M: 1-by-K matrix where M(i) is the max power of 2 such that 2^M(i) divides n function [m, M] = Omega2(n)   M = NaN*zeros(1, n);   M(1)=0; M(2)=1;     for k=3:n       if M(k-2)~=0         M(k)=M(k-k/2)+1;       else         M(k)=0;       end     end     m=M(end); end % Redjan Shabani, July 17 2012 CROSSREFS A053398(1,n). Column/row 1 of table A050602. First differences of A011371. Bisection of A050605 and |A088705|. Cf. A122840, A122841, A007949 (3-adic), A112765 (5-adic), A214411 (7-adic). See A050603 and A136480 for a(n) + a(n+1). This is Guy Steele's sequence GS(1, 4) (see A135416). Cf. A006519, A002487, A063787, A000079, A051064, A055457, A220466. Sequence in context: A162590 A191258 A191255 * A225345 A083280 A060689 Adjacent sequences:  A007811 A007812 A007813 * A007815 A007816 A007817 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Formula index adapted to the offset of A025480 - R. J. Mathar, Jul 20 2010 STATUS approved

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