A130068 Maximal power of 2 dividing the binomial coefficient binomial(m, 2^k) where m >= 1 and 1 <= 2^k <= m.

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

Offset: 1

Author

Hieronymus Fischer, May 05 2007, Sep 10 2007

Keywords

• nonn
• tabl

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A130067, A065040, A001855, A030308.

Formula

a(n)=g(m)-g(m-2^k) where g(x)=sum(floor(x/2^i), kA001855(j)A001855(m). Also true: a(n)=sum(product(1-b(i), k<=i

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