

A070940


Number of digits that must be counted from left to right to reach the last 1 in the binary representation of n.


9



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, 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, 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
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OFFSET

1,3


COMMENTS

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, Jan 26 2003
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,..,n1. Proof using Kummer's Theorem given by Marc Schwartz.  Labos Elemer, Apr 23 2003
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 Elemer
Smallest x>0 for which a(x)=n equals 2^n.  Labos Elemer
a(n) <= A070939(n), a(n) = A070939(n) iff n is odd, where A070939(n) = floor(log_2(n)) + 1.  Reinhard Zumkeller, Jan 26 2003
Can be regarded as a table with row n having 2^(n1) columns, with odd columns repeating the previous row, and even columns containing the row number.  Franklin T. AdamsWatters, Nov 08 2011
It appears that a(n) is the greatest number in a periodicity equivalence class defined at A269570; e.g., the 5 classes for n = 35 are (1, 1, 2, 2, 6), (1, 1, 1, 1, 4, 2, 2), (3), (1, 3), (1, 2); in these the greatest number is 6, so that a(35) = 6.  Clark Kimberling, Mar 01 2016


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n


FORMULA

a(n) = [log2(n)]  A007814(n) = A070939(n)  A007814(n).
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, Feb 01 2003
G.f.: sum(k>=0, t/(1t^2) * [1 + sum(l>=1, t^2^l)], t=x^2^k).  Ralf Stephan, Mar 15 2004


EXAMPLE

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.
The table starts:
1
1 2
1 3 2 3
1 4 3 4 2 4 3 4


MAPLE

A070940 := n > if n mod 2 = 0 then A070939(n)A001511(n/2) else A070939(n); fi;


MATHEMATICA

Table[Length[Union[Table[GCD[2^n, Binomial[n, j]], {j, 0, n}]]], {n, 0, 256}]
f[n_] := Position[ IntegerDigits[n, 2], 1][[ 1, 1]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Dec 01 2004 *)


PROG

(Haskell)
a070940 = maximum . a080080_row  Reinhard Zumkeller, Apr 22 2013


CROSSREFS

Cf. A070939, A001511. Differs from A002487 around 11th term.
Cf. A000005, A007318, A000079, A082907, A082908.
Bisections give A070941 and this sequence (again).
Cf. A002064 (row sums), A199570.
Sequence in context: A215467 A038568 A071912 * A020651 A002487 A263017
Adjacent sequences: A070937 A070938 A070939 * A070941 A070942 A070943


KEYWORD

nonn,nice,easy,tabf


AUTHOR

N. J. A. Sloane, May 18 2002


EXTENSIONS

Entry revised by Ralf Stephan, Nov 29 2004


STATUS

approved



