Search: id:A070940
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%I A070940
%S A070940 1,1,2,1,3,2,3,1,4,3,4,2,4,3,4,1,5,4,5,3,5,4,5,2,5,4,5,3,5,4,5,
%T A070940 1,6,5,6,4,6,5,6,3,6,5,6,4,6,5,6,2,6,5,6,4,6,5,6,3,6,5,6,4,6,5,
%U A070940 6,1,7,6,7,5,7,6,7,4,7,6,7,5,7,6,7,3,7,6,7,5,7,6,7,4,7,6,7,5,7
%N A070940 Number of digits that must be counted from left to right to reach the 
               last 1 in the binary representation of n.
%C A070940 Length of longest carry sequence when adding numbers <= n to n in binary 
               representation: a(n)=T(n, A080079(n)) and T(n,k)<=a(n) for 1<=k<=n, 
               with T defined as in A080080. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jan 26 2003
%C A070940 a(n+1) is the number of distinct values of GCD[2^n,C[n,j]] (or, equivalently, 
               A007814(C(n,j))) arising if j=0,..,n-1. Proof using Kummer's Theorem 
               given by Marc Schwartz. - Labos E., Apr 23, 2003
%C A070940 E.g. n=10: 10th row of Pascal's triangle = {1,10,45,120,210,252,210,120,
               45,10,1}, largest powers of 2 dividing binomial coefficients is: 
               {1,2,1,8,2,4,2,8,1,2,1}; including distinct powers of 2, thus a(10)=4. 
               If m=-1+2^k, i.e. m=0,1,3,7,15,31,.. then a(m)=1. This corresponds 
               to "odd rows" of Pascal triangle. (Labos)
%C A070940 Smallest x>0 for which a(x)=n equals 2^n. (Labos)
%C A070940 a(n) <= A070939(n), a(n) = A070939(n) iff n is odd, where A070939(n) 
               = floor(log_2(n)) + 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jan 26 2003
%H A070940 <a href="Sindx_Bi.html#binary">Index entries for sequences related to 
               binary expansion of n</a>
%F A070940 a(n) = [log2(n)] - A007814(n) = A070939(n) - A007814(n).
%F A070940 a(n) = f(n, 1), f(n, k) = if n=1 then k else f(floor(n/2), k+(if k>1 
               then 1 else n mod 2)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 01 2003
%F A070940 G.f.: sum(k>=0, t/(1-t^2) * [1 + sum(l>=1, t^2^l)], t=x^2^k). - R. Stephan, 
               Mar 15 2004
%e A070940 a(10)=3 is the number of digits that must be counted from left to right 
               to reach the last 1 in 1010, the binary representation of 10.
%p A070940 A070940 := n -> if n mod 2 = 0 then A070939(n)-A001511(n/2) else A070939(n); 
               fi;
%t A070940 Table[Length[Union[Table[GCD[2^n, Binomial[n, j]], {j, 0, n}]]], {n, 
               0, 256}]
%t A070940 f[n_] := Position[ IntegerDigits[n, 2], 1][[ -1, 1]]; Table[ f[n], {n, 
               105}] (from Robert G. Wilson v Dec 01 2004)
%Y A070940 Cf. A070939, A001511. Differs from A002487 around 11-th term.
%Y A070940 Cf. A000005, A007318, A000079, A082907, A082908.
%Y A070940 Bisections give A070941 and this sequence (again).
%Y A070940 Sequence in context: A057432 A038568 A071912 this_sequence A020651 A160232 
               A002487
%Y A070940 Adjacent sequences: A070937 A070938 A070939 this_sequence A070941 A070942 
               A070943
%K A070940 nonn,nice,easy
%O A070940 1,3
%A A070940 N. J. A. Sloane (njas(AT)research.att.com), May 18 2002
%E A070940 Entry revised by Ralf Stephan, Nov 29 2004

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