A359359 Sum of positions of zeros in the binary expansion of n, where positions are read starting with 1 from the left (big-endian).

1, 0, 2, 0, 5, 2, 3, 0, 9, 5, 6, 2, 7, 3, 4, 0, 14, 9, 10, 5, 11, 6, 7, 2, 12, 7, 8, 3, 9, 4, 5, 0, 20, 14, 15, 9, 16, 10, 11, 5, 17, 11, 12, 6, 13, 7, 8, 2, 18, 12, 13, 7, 14, 8, 9, 3, 15, 9, 10, 4, 11, 5, 6, 0, 27, 20, 21, 14, 22, 15, 16, 9, 23, 16, 17, 10

Offset: 0

Author

Gus Wiseman, Jan 03 2023

Keywords

• nonn
• base

Examples

			The binary expansion of 100 is (1,1,0,0,1,0,0), with zeros at positions {3,4,6,7}, so a(100) = 20.
		

Crossrefs

The number of zeros is A023416, partial sums A059015.

For positions of 1's we have A230877, reversed A029931.

The reversed version is A359400.

A003714 lists numbers with no successive binary indices.

A030190 gives binary expansion.

A039004 lists the positions of zeros in A345927.

Cf. A000120, A048793, A065359, A069010, A070939, A073642, A083652, A328594, A328595, A359402, A359495.

Programs

Mathematica

Table[Total[Join@@Position[IntegerDigits[n,2],0]],{n,0,100}]

Formula

a(n>0) = binomial(A029837(n)+1,2) - A230877(n).