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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 (list; graph; refs; listen; history; text; internal format)
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,..,n-1. Proof using Kummer's Theorem given by Marc Schwartz. - Labos E., 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)

Smallest x>0 for which a(x)=n equals 2^n. (Labos)

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^(n-1) columns, with odd columns repeating the previous row, and even columns containing the row number. - Franklin T. Adams-Watters, Nov 08 2011.

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/(1-t^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}] (from 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 11-th 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 A060162

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

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Last modified April 16 12:16 EDT 2014. Contains 240591 sequences.