A089633 Numbers having no more than one 0 in their binary representation.

0, 1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 23, 27, 29, 30, 31, 47, 55, 59, 61, 62, 63, 95, 111, 119, 123, 125, 126, 127, 191, 223, 239, 247, 251, 253, 254, 255, 383, 447, 479, 495, 503, 507, 509, 510, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1023

Offset

0,3

Comments

Complement of A158582. - Reinhard Zumkeller, Apr 16 2009

Also union of A168604 and A030130. - Douglas Latimer, Jul 19 2012

Numbers of the form 2^t - 2^k - 1, 0 <= k < t.

n is in the sequence if and only if 2*n+1 is in the sequence. - Robert Israel, Dec 14 2018

Also the least binary rank of a strict integer partition of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 24 2024

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv:0801.0072 [math.CO], 2007-201. See Section 14.

Vladimir Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8.

Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1.

Index entries for sequences related to binary expansion of n.

Formula

A023416(a(n)) <= 1; A023416(a(n)) = A023532(n-2) for n>1;

A000120(a(u)) <= A000120(a(v)) for u<v; A000120(a(n)) = A003056(n).

a(0)=0, n>0: a(n+1) = Min{m>n: BinOnes(a(n))<=BinOnes(m)} with BinOnes=A000120.

If m = floor((sqrt(8*n+1) - 1) / 2), then a(n) = 2^(m+1) - 2^(m*(m+3)/2 - n) - 1. - Carl R. White, Feb 10 2009

A029931(a(n)) = n and A029931(m) != n for m < a(n). - Reinhard Zumkeller, Feb 28 2014

A265705(a(n),k) = A265705(a(n),a(n)-k), k = 0 .. a(n). - Reinhard Zumkeller, Dec 15 2015

a(A014132(n)-1) = 2*a(n-1)+1 for n >= 1. - Robert Israel, Dec 14 2018

Sum_{n>=1} 1/a(n) = A065442 + A160502 = 3.069285887459... . - Amiram Eldar, Jan 09 2024

A019565(a(n)) = A077011(n). - Gus Wiseman, May 24 2024

Example

From Tilman Piesk, May 09 2012: (Start)
This may also be viewed as a triangle:             In binary:
                  0                                         0
               1     2                                 01       10
             3    5    6                          011      101      110
           7   11   13   14                  0111     1011     1101     1110
        15   23   27   29   30          01111    10111    11011    11101    11110
      31  47   55   59   61   62
   63   95  111  119  123  125  126
Left three diagonals are A000225,  A055010, A086224. Right diagonal is A000918. Central column is A129868. Numbers in row n (counted from 0) have n binary 1s. (End)
From Gus Wiseman, May 24 2024: (Start)
The terms together with their binary expansions and binary indices begin:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
  11:   1011 ~ {1,2,4}
  13:   1101 ~ {1,3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  23:  10111 ~ {1,2,3,5}
  27:  11011 ~ {1,2,4,5}
  29:  11101 ~ {1,3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  47: 101111 ~ {1,2,3,4,6}
  55: 110111 ~ {1,2,3,5,6}
  59: 111011 ~ {1,2,4,5,6}
  61: 111101 ~ {1,3,4,5,6}
  62: 111110 ~ {2,3,4,5,6}
(End)

Maple

seq(seq(2^a-1-2^b, b=a-1..0, -1), a=1..11); # Robert Israel, Dec 14 2018

Mathematica

fQ[n_] := DigitCount[n, 2, 0] < 2; Select[ Range[0, 2^10], fQ] (* Robert G. Wilson v, Aug 02 2012 *)

Prog

(Haskell)
a089633 n = a089633_list !! (n-1)
a089633_list = [2 ^ t - 2 ^ k - 1 | t <- [1..], k <- [t-1, t-2..0]]
-- Reinhard Zumkeller, Feb 23 2012
(PARI) {insq(n) = local(dd, hf, v); v=binary(n); hf=length(v); dd=sum(i=1, hf, v[i]); if(dd<=hf-2, -1, 1)}
{for(w=0, 1536, if(insq(w)>=0, print1(w, ", ")))}
\\ Douglas Latimer, May 07 2013
(PARI) isoka(n) = #select(x->(x==0), binary(n)) <= 1; \\ Michel Marcus, Dec 14 2018
(Python)
from itertools import count, islice
def A089633_gen(): # generator of terms
    return ((1<<t)-(1<<k)-1 for t in count(1) for k in range(t-1, -1, -1))
A089633_list = list(islice(A089633_gen(), 30)) # Chai Wah Wu, Feb 10 2023
(Python)
from math import isqrt, comb
def A089633(n): return (1<<(a:=(isqrt((n<<3)+1)-1>>1)+1))-(1<<comb(a, 2)+a-1-n)-1 # Chai Wah Wu, Dec 19 2024

Crossrefs

Cf. A000120, A007088, A011371, A014132, A023416, A023532, A029931.

Cf. A181741 (primes), union of A081118 and A000918, apart from initial -1.

Cf. A065442, A160502, A265705.

For least binary index (instead of rank) we have A001511.

Applying A019565 (Heinz number of binary indices) gives A077011.

For greatest binary index we have A029837 or A070939, opposite A070940.

Row minima of A118462 (binary ranks of strict partitions).

For sum instead of minimum we have A372888, non-strict A372890.

A000009 counts strict partitions, ranks A005117.

A048675 gives binary rank of prime indices, distinct A087207.

A048793 lists binary indices, product A096111, reverse A272020.

A277905 groups all positive integers by binary rank of prime indices.

Cf. A000041, A005940, A018819, A147655, A225620, A246868.

Sequence in context: A342392 A053328 A333786 * A362815 A358772 A003172

Adjacent sequences: A089630 A089631 A089632 * A089634 A089635 A089636

Keyword

nonn,base

Author

Reinhard Zumkeller, Jan 01 2004

Status

approved