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

1,2

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

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

Example

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

Mathematica

Select[Range[130], DigitCount[#, 2, 0]<DigitCount[#, 2, 1]&]  (* Harvey P. Dale, Jan 12 2011 *)

Prog

(Haskell)
a072600 n = a072600_list !! (n-1)
a072600_list = filter ((< 0) . a037861) [0..]
-- Reinhard Zumkeller, Mar 31 2015
(PARI) is(n)=2*hammingweight(n)>exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

Crossrefs

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

Sequence in context: A328557 A216782 A328952 * A047582 A333217 A342438

Adjacent sequences: A072597 A072598 A072599 * A072601 A072602 A072603

Keyword

nonn,base,easy

Author

Reinhard Zumkeller, Jun 23 2002

Status

approved