A072600 Numbers which in base 2 have fewer 0's than 1's.

1, 3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 39, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115

Offset: 1

Author

Reinhard Zumkeller, Jun 23 2002

Keywords

• nonn
• base
• easy

Comments

A037861(a(n)) < 0.

b_k = {a(n) | for all n s.t. a(n) contains k binary digits equal to 1} is the list of all valid win/loss round sequences in a "best of 2k-1" two player game, where 1 is a win and 0 is a loss. For example 19 = 10011b represents a game where the winner won the first two rounds, lost the next two, and won the last one. |b_k| = A001700(k). - Philippe Beaudoin, May 14 2014

Examples

			11 is present because '1011' contains 1 '0' and 3 '1's: 1<3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..4733 ( numbers < 2^13)
• Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
• Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A007088, A000120, A023416, A072601, A031443, A072602, A072603, A037861

Programs

Haskell

a072600 n = a072600_list !! (n-1)
a072600_list = filter ((< 0) . a037861) [0..]
-- Reinhard Zumkeller, Mar 31 2015

Mathematica

Select[Range[130],DigitCount[#,2,0]Harvey P. Dale, Jan 12 2011 *)

PARI

is(n)=2*hammingweight(n)>exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020