A036993 -id:A036993 - cp's OEIS frontend

A036994 Numbers k with the property that reading from right to left in the binary expansion of k, the number of 1's always stays ahead of the number of 0's.

1, 3, 7, 11, 15, 23, 27, 31, 39, 43, 47, 55, 59, 63, 79, 87, 91, 95, 103, 107, 111, 119, 123, 127, 143, 151, 155, 159, 167, 171, 175, 183, 187, 191, 207, 215, 219, 223, 231, 235, 239, 247, 251, 255, 287, 303, 311, 315, 319, 335, 343, 347, 351, 359, 363, 367

Offset: 1

N. J. A. Sloane

Even numbers can't appear in this sequence. - Alonso del Arte, Sep 21 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A036990, A036991, A036992, A036993.

Cf. A030308.

a036994 n = a036994_list !! (n-1)
a036994_list = filter ((p 0) . a030308_row) [1, 3 ..] where
   p ones []    = ones > 0
   p ones (0:bs) = ones > 1 && p (ones - 1) bs
   p ones (1:bs) = p (ones + 1) bs
-- Reinhard Zumkeller, Aug 01 2013

aheadOnesRLQ[n_Integer] := Module[{digits, len, flag = True, iter = 1, ones = 0, zeros = 0}, digits = Reverse[IntegerDigits[n, 2]]; len = Length[digits]; While[flag && iter < len, If[digits[[iter]] == 1, ones++, zeros++]; flag = ones > zeros; iter++]; flag]; Select[Range[1, 201, 2], aheadOnesRLQ] (* Alonso del Arte, Sep 21 2011 *)
Select[Range[400],Min[Accumulate[Reverse[IntegerDigits[#,2]/. (0->-1)]]]> 0&] (* Harvey P. Dale, Apr 23 2016 *)

ok(x)={if(x<1,return(0));my(c=logint(x,2),c0=0,c1=0);for(i=0,c,if(bittest(x,i),c1++,c0++);if(c1<=c0,return(0)));1} \\ for(n=1,367,if(ok(n),print1(n,", "))) - Ruud H.G. van Tol, Sep 14 2022

from itertools import count, islice
def A036994_gen(startvalue=0): # generator of terms >= startvalue
    for n in count(max(startvalue,0)):
        s = bin(n)[2:]
        c, l = 0, len(s)
        for i in range(l):
            c += int(s[l-i-1])
            if 2*c <= i + 1:
                break
        else:
            yield n
A036994_list = list(islice(A036994_gen(),20)) # Chai Wah Wu, Dec 31 2021

More terms from Erich Friedman

A343049 The k-th binary digit of a(n) is the most frequent digit among the first k binary digits of n (in case of a tie, take the k-th binary digit of n).

Original entry on oeis.org

0, 1, 2, 7, 0, 5, 6, 31, 0, 9, 10, 31, 8, 29, 30, 127, 0, 1, 2, 23, 0, 21, 22, 127, 0, 25, 26, 127, 24, 125, 126, 511, 0, 1, 2, 39, 0, 37, 38, 127, 0, 41, 42, 127, 40, 125, 126, 511, 0, 33, 34, 119, 32, 117, 118, 511, 32, 121, 122, 511, 120, 509, 510, 2047, 0

Offset: 0

Rémy Sigrist, Apr 09 2021

Leading zeros are taken into account up to the point the number of zeros exceeds the total number of ones.

We scan the binary representation of a number starting from the least significant digit. See A343271 for the other way.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     7      11        111
   4     0     100          0
   5     5     101        101
   6     6     110        110
   7    31     111      11111
   8     0    1000          0
   9     9    1001       1001
  10    10    1010       1010
  11    31    1011      11111
  12     8    1100       1000
  13    29    1101      11101
  14    30    1110      11110
  15   127    1111    1111111

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A032925, A036993, A070939, A343271, A342697.

a(n, base=2) = { my (d=digits(n, base), t, f=vector(base)); d=concat(vector(#d), d); forstep (k=#d, 1, -1, f[1+d[k]]++; if (vecmax(f)==f[1+d[k]], t=d[k];); d[k]=t); fromdigits(d, base) }

a(n) = 0 iff n belongs to A036993.
a(n) = n iff n = 0 or n belongs to A032925.
a(2^k-1) = 2^(2*k-1)-1 for any k > 1.
A070939(a(n)) < 2*A070939(n).

cp's OEIS Frontend

A036994 Numbers k with the property that reading from right to left in the binary expansion of k, the number of 1's always stays ahead of the number of 0's.

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