A326669 Numbers k such that the average position of the ones in the binary expansion of k is an integer.

1, 2, 4, 5, 7, 8, 10, 14, 16, 17, 20, 21, 27, 28, 31, 32, 34, 35, 39, 40, 42, 49, 54, 56, 57, 62, 64, 65, 68, 70, 73, 78, 80, 84, 85, 93, 98, 99, 107, 108, 112, 114, 119, 124, 127, 128, 130, 133, 136, 140, 141, 146, 147, 155, 156, 160, 161, 167, 168, 170, 175

Offset: 1

Author

Gus Wiseman, Jul 17 2019

Keywords

• nonn
• base

Comments

These are numbers whose exponents in their representation as a sum of distinct powers of 2 have integer average.

Examples

			42 is in the sequence because 42 = 2^1 + 2^3 + 2^5 and the average of {1,3,5} is 3, an integer.
		

Crossrefs

Cf. A000120, A051293, A070939, A102627, A291165, A291166, A316413, A326622, A326668, A326672, A326673.

Programs

Mathematica

Select[Range[100],IntegerQ[Mean[Join@@Position[IntegerDigits[#,2],1]]]&]

PARI

isok(m) = my(b=binary(m)); denominator(vecsum(Vec(select(x->(x==1), b, 1)))/hammingweight(m)) == 1; \\ Michel Marcus, Jul 02 2021