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Search: id:A130068
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| A130068 |
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Maximal power of 2 dividing the binomial coefficient binomial(m,2^k) where m>=1 and 1<=2^k<=m. |
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+0
2
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| 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 3, 2, 1, 0, 1, 0, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 3, 2, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0
(list; table; graph; listen)
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OFFSET
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1,6
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COMMENT
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Provided m and k are given, the sequence index n is n=A001855(m)+k+1. Ordered by m as rows and k as columns the sequence forms a sort of a logarithmically distorted triangle.
a(n) is the maximal power of 2 dividing A130067(n).
Equivalent propositions: (1) a(n)=0; (2) A130067(n) is odd; (3) the k-th digit of m is 1; (4) A030308(n)=1.
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FORMULA
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a(n)=g(m)-g(m-2^k) where g(x)=sum(floor(x/2^i), k<i<=floor(log_2(x))), m=max(j | A001855(j)<n) and k=n-1-A001855(m). Also true: a(n)=sum(product(1-b(i), k<=i<j), k<j<=floor(log_2(m))) where b(i) is the i-th digit of the binary representation of m. Example: n=35 gives m=12 and k=1, so that m=1100 and a(n)=1-b(1)+(1-b(1))(1-b(2))=1+0=1.
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EXAMPLE
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a(6)=2 since 2^2 divides binomial(4,2^0)=4 and 2^3 is not a factor (here n=6 gives m=4, k=0).
a(20)=1 since 2^1 divides binomial(8,2^2)=70 and 2^2 is not a factor (here n=20 gives m=8, k=2).
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CROSSREFS
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Cf. A130067, A065040, A001855, A030308.
Sequence in context: A122855 A140727 A140728 this_sequence A051699 A007920 A127587
Adjacent sequences: A130065 A130066 A130067 this_sequence A130069 A130070 A130071
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KEYWORD
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nonn,tabl
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AUTHOR
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Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 05 2007, Sep 10 2007
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