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A072600 Numbers which in base 2 have fewer 0's than 1's. 12
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 (list; graph; refs; listen; history; text; internal format)
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.
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
Sequence in context: A328557 A216782 A328952 * A047582 A333217 A342438
KEYWORD
nonn,base,easy
AUTHOR
Reinhard Zumkeller, Jun 23 2002
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)