A030130 -id:A030130 - cp's OEIS frontend

A055010 a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.

0, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Henry Bottomley, May 31 2000

Apart from leading term (which should really be 3/2), same as A083329.
Written in binary, a(n) is 1011111...1.
The sequence 2, 5, 11, 23, 47, 95, ... apparently gives values of n such that Nim-factorial(n) = 2. Cf. A059970. However, compare A060152. More work is needed! - John W. Layman, Mar 09 2001
With offset 1, number of (132,3412)-avoiding two-stack sortable permutations.
Number of descents after n+1 iterations of morphism A007413.
a(n) = A164874(n,1), n>0; subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-1). - Milan Janjic, Jan 24 2010
a(n) is the total number of records over all length n binary words. A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all iGeoffrey Critzer, Jul 18 2020
Called Thabit numbers after the Syrian mathematician Thābit ibn Qurra (826 or 836 - 901). - Amiram Eldar, Jun 08 2021
a(n) is the number of objects in a pile that represents a losing position in a Nim game, where a player must select at least one object but not more than half of the remaining objects, on their turn. - Kiran Ananthpur Bacche, Feb 03 2025

			a(3) = 3*2^2 - 1 = 3*4 - 1 = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric S. Egge and Toufik Mansour, 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, arXiv:math/0205206 [math.CO], 2002.
Sergey Kitaev and Toufik Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Hunar Sherzad Taher and Saroj Kumar Dash, On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 448-459. See p. 2.
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number.
Wikipedia, Thabit number.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A007505 for primes in this sequence. Apart from initial term, same as A052940 and A083329.
Cf. A266550 (independence number of the n-Mycielski graph).
Cf. A000079, A000225, A007283, A007413, A010036, A030130, A033484, A059970, A060152, A099258, A118654, A164874, A196168.

Concatenation([0], List([1..35], n-> 3*2^(n-1)-1)); # G. C. Greubel, May 06 2019

[Floor(3*2^(n-1) - 1): n in [0..35]]; // Vincenzo Librandi, May 18 2011

Join[{0},3*2^Range[0,34]-1] (* Harvey P. Dale, May 05 2013 *)

a(n)=3*2^n\2 - 1 \\ Charles R Greathouse IV, Apr 08 2016

[0]+[3*2^(n-1)-1 for n in (1..35)] # G. C. Greubel, May 06 2019

a(n) = A118654(n-1, 4), for n > 0.
a(n) = 2*a(n-1) + 1 = a(n-1) + A007283(n-1) = A007283(n)-1 = A000079(n) + A000225(n + 1) = A000079(n + 1) + A000225(n) = 3*A000079(n) - 1 = 3*A000225(n) + 2.
a(n) = A010036(n)/2^(n-1). - Philippe Deléham, Feb 20 2004
a(n) = A099258(A033484(n)-1) = floor(A033484(n)/2). - Reinhard Zumkeller, Oct 09 2004
G.f.: x*(2-x)/((1-x)*(1-2*x)). - Philippe Deléham, Oct 04 2011
a(n+1) = A196168(A000079(n)). - Reinhard Zumkeller, Oct 28 2011
E.g.f.: (3*exp(2*x) - 2*exp(x) - 1)/2. - Stefano Spezia, Sep 14 2024

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

Original entry on oeis.org

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

Reinhard Zumkeller, Jan 01 2004

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

			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)

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.

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.

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

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

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

{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

isoka(n) = #select(x->(x==0), binary(n)) <= 1; \\ Michel Marcus, Dec 14 2018

from itertools import count, islice
def A089633_gen(): # generator of terms
    return ((1<A089633_list = list(islice(A089633_gen(),30)) # Chai Wah Wu, Feb 10 2023

from math import isqrt, comb
def A089633(n): return (1<<(a:=(isqrt((n<<3)+1)-1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

A023416(a(n)) <= 1; A023416(a(n)) = A023532(n-2) for n>1;
A000120(a(u)) <= A000120(a(v)) for uA000120(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

A086224 a(n) = 7*2^n - 1.

Original entry on oeis.org

6, 13, 27, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511, 234881023, 469762047, 939524095, 1879048191, 3758096383, 7516192767, 15032385535, 30064771071

Offset: 0

Marco Matosic, Jul 27 2003

a(n) = A164874(n+2,2); subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-5). - Milan Janjic, Jan 27 2010

Michael De Vlieger, Table of n, a(n) for n = 0..3319
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-5, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Other sequences with recurrence a(n+1) = 2*a(n) + 1 are:
a(0) = 2 gives A153893, a(0)=3 essentially A126646.
a(0) = 4 gives A153894, a(0)=5 essentially A153893.
a(0) = 7 gives essentially A000225.
a(0) = 8 gives A052996 except for some initial terms,
a(0) = 9 is essentially A153894.
a(0) = 10 gives A086225,
a(0) = 11 is essentially A153893.
a(0) = 13 is essentially A086224.
Cf. A030130, A052940, A164874.

7*2^Range[0,30]-1 (* Harvey P. Dale, May 09 2018 *)

a(n)=7<Charles R Greathouse IV, Sep 24 2015

a(n+1) = 2*a(n) + 1.
G.f.: (6-5*x)/((1-x)*(1-2*x)). - Jaume Oliver Lafont, Sep 14 2009
a(n-1)^2 + a(n) = (7*2^(n-1))^2. - Vincenzo Librandi, Aug 08 2010
a(n) = (A052940(n+1) + A000225(n+3))/2. - Gennady Eremin, Aug 31 2023
From Elmo R. Oliveira, Apr 22 2025: (Start)
E.g.f.: exp(x)*(7*exp(x) - 1).
a(n) = 3*a(n-1) - 2*a(n-2). (End)

More terms from David Wasserman, Feb 22 2005

A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

Original entry on oeis.org

2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046

Offset: 1

Reinhard Zumkeller, Aug 29 2009

All terms contain exactly 1 zero in binary representation.

			Initial rows:
   1:                             2
   2:                        5        6
   3:                  11        13        14
   4:             23        27       29        30
   5:        47        55        59        61        62
   6:    95       111       119      123       125       126
also in binary representation:
                                 10
                            101       110
                      1011      1101      1110
                 10111     11011     11101     11110
           101111    110111    111011    111101    111110
      1011111   1101111   1110111   1111011   1111101   1111110 .

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A030130, A023416, A059673, A055010, A086224, A036563, A000918.

a164874 n k = a164874_tabl !! (n-1) !! (k-1)
a164874_row n = a164874_tabl !! (n-1)
a164874_tabl = map reverse $ iterate f [2] where
   f xs@(x:_) = (2 * x + 2) : map ((+ 1) . (* 2)) xs
-- Reinhard Zumkeller, Mar 31 2015

A164874row[n_] := 2^(n + 1) - 1 - BitShiftRight[2^n, Range[n]];
Array[A164874row, 10] (* Paolo Xausa, Jun 13 2025 *)

from math import isqrt
def A164874(n): return (1<<(a:=(isqrt(n<<3)+1>>1)+1))-(1<<(a*(a-1)>>1)-n)-1 # Chai Wah Wu, May 21 2025

T(n,k) = 2^(n+1) - 2^(n-k) - 1, 1 <= k <= n.
T(n,k) = A030130(n*(n-1)/2 + k + 1);
A023416(T(n,k)) = 1, 1<=k<=n;
A059673(n) = sum of n-th row;
T(n,1) = A055010(n);
T(n,2) = A086224(n-2) for n > 1;
T(n,n-1) = A036563(n+1) for n > 1;
T(n,n) = A000918(n+1).

A095078 Primes with a single 0 bit in their binary expansion.

Original entry on oeis.org

2, 5, 11, 13, 23, 29, 47, 59, 61, 191, 223, 239, 251, 383, 479, 503, 509, 991, 1019, 1021, 2039, 3583, 3967, 4079, 4091, 4093, 6143, 15359, 16127, 16319, 16381, 63487, 65407, 65519, 129023, 131063, 245759, 253951, 261631, 261887, 262079

Offset: 1

Antti Karttunen, Jun 01 2004

Except for the first value 2, the sequence gives the primes of the form 2^k -2^j -1 with 0 < j < k-1. If j=k-1 we obtain the Mersenne primes. - Pierre CAMI, May 19 2005

{2} UNION (A000040 INTERSECT A190620). - Chai Wah Wu, Dec 19 2024

T. D. Noe, Table of n, a(n) for n=1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A030130. Cf. A095058, A190620.

Select[Prime[Range[23000]],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Nov 28 2019 *)

forprime(p=2,262079,v=binary(p);s=0;for(k=1,#v,s+=v[k]);if(#v-s==1,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

A059673 Sum of binary numbers with n 1's and one (non-leading) 0.

Original entry on oeis.org

0, 2, 11, 38, 109, 284, 699, 1658, 3833, 8696, 19447, 42998, 94197, 204788, 442355, 950258, 2031601, 4325360, 9175023, 19398638, 40894445, 85983212, 180355051, 377487338, 788529129, 1644167144, 3422552039, 7113539558, 14763950053

Offset: 0

Henry Bottomley, Feb 05 2001

For n>0, a(n) = sum of n-th row of the triangle in A164874. [Reinhard Zumkeller, Aug 29 2009]

			a(4)=109 since the binary sum 11110+11101+11011+10111 is 30+29+27+23.

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A030130, A164874. [From Reinhard Zumkeller, Aug 29 2009]

A059673:=n->(2*n-1)*2^n+1-n: seq(A059673(n), n=0..50); # Wesley Ivan Hurt, Apr 24 2017

Table[Total[FromDigits[#,2]&/@Rest[Permutations[PadRight[{0},n,1]]]],{n,30}] (* or *) LinearRecurrence[{6,-13,12,-4},{0,2,11,38},30] (* Harvey P. Dale, May 17 2015 *)

concat(0, Vec(-x*(2*x^2+x-2)/((x-1)^2*(2*x-1)^2) + O(x^100))) \\ Colin Barker, Sep 14 2014

def A059673(n): return ((n<<1)-1<Chai Wah Wu, Dec 19 2024

a(n) = (2n-1)*2^n+1-n.
G.f.: -x*(2*x^2+x-2) / ((x-1)^2*(2*x-1)^2). - Colin Barker, Sep 14 2014
a(0)=0, a(1)=2, a(2)=11, a(3)=38, a(n)=6*a(n-1)-13*a(n-2)+ 12*a(n-3)- 4*a(n-4). - Harvey P. Dale, May 17 2015. [This is equivalent to the g.f. -x*(2*x^2+x-2) / ((x-1)^2*(2*x-1)^2) given by Colin Barker. - N. J. A. Sloane, May 17 2015]

A160502 Decimal expansion of the (finite) value of Sum_{ k >= 1, k has only a single zero digit in base 2 } 1/k.

Original entry on oeis.org

1, 4, 6, 2, 5, 9, 0, 7, 3, 5, 0, 4, 4, 3, 6, 4, 6, 9, 9, 5, 4, 6, 1, 4, 5, 4, 4, 6, 7, 2, 0, 5, 3, 4, 6, 2, 1, 0, 7, 4, 7, 4, 4, 8, 6, 4, 7, 4, 8, 8, 2, 1, 1, 0, 9, 3, 6, 4, 2, 0, 0, 6, 2, 4, 3, 5, 4, 5, 2, 2, 9, 4, 3, 7, 8, 5, 8, 8, 1, 5, 0, 3, 5, 5, 2, 1, 9, 2, 9, 2, 2, 1, 5, 9, 2, 4, 0, 8, 9, 2, 3, 6, 9, 7, 5

Offset: 1

Robert G. Wilson v, May 15 2009

The sum of 1/n where n has a single 0 in base 2.

			1.4625907350443646995461454467205346210747448647488211093642006243545229...

Robert Baillie, Summing The Curious Series Of Kempner And Irwin, arXiv:0806.4410 [math.CA], 2008-2015.

Cf. A030130 (numbers with a single zero in base 2), A140502.

RealDigits[ N[ Sum[1/(2^n - 1 - 2^k), {n, 2, 400}, {k, 0, n - 2}], 111]][[1]]
(* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[0, 1, 111, 2] (* Robert G. Wilson v, Aug 03 2010 *)

Equals Sum_{n>=2} Sum_{k=0..n-2}, 1/(2^n - 1 - 2^k).

A383666 Numbers in whose binary representation no bit (0 or 1) occurs exactly once.

Original entry on oeis.org

3, 7, 9, 10, 12, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Clark Kimberling, May 07 2025

Also numbers that are not a power of 2 and are not (2^k + 1) away from the next larger power of 2 for some k. - David A. Corneth, May 17 2025

			From _David A. Corneth_, May 17 2025: (Start)
3 = 11_2 is in the sequence as both the digits 0 and the digits 1 do not occur exactly once in the binary expansion. Also 3 is no power of 2 and one less than a power of 2.
6 = 101_2 is not in the sequence as the digit 0 occurs exactly once in the binary expansion. Also it can be written as 2^3 - 2^0 - 1. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A030130, A158581, A383667.

filter:= proc(n) local L,n1,n0;
  L:= convert(n,base,2);
  n1:= convert(L,`+`);
  n0:= nops(L)-n1;
  n1 >= 2 and n0 <> 1
end proc;
select(filter, [$1..1000]); # Robert Israel, May 13 2025

s = Select[Range[200], DigitCount[#, 2, 0] != 1 && DigitCount[#, 2, 1] != 1 &]
Map[First, RealDigits[s, 2]]

isok(k) = my(b=binary(k)); (#select(x->(x==1), b) != 1) && (#select(x->(x==0), b) != 1); \\ Michel Marcus, May 13 2025

is(n) = {
    my(v = valuation(n, 2));
    if(n >> v == 1, return(0));
    if(1<> valuation(c, 2) == 1, return(0));
    1
} \\ David A. Corneth, May 17 2025

upto(n) = {
    my(res = [1..n], del = List());
    for(i = 0, logint(n, 2)+1,
        pow2 = 1<David A. Corneth, May 17 2025

def A383666(n):
    def f(x):
        if x<=1: return n+x
        l, s = x.bit_length(), bin(x)[2:]
        if (m:=s.count('0'))>0: return n+s.index('0')-(m>1)+(l*(l-1)>>1)
        return n-1+(l*(l+1)>>1)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, May 21 2025

A214853 Fibonacci numbers with only one 0 in the binary representation.

Original entry on oeis.org

0, 2, 5, 13, 55

Offset: 1

Alex Ratushnyak, Mar 08 2013

Conjecture: the sequence is finite.
No more terms below 2*10^301. - Matthew House, Sep 06 2015
No more terms below 10^162809483. (This number could easily be raised. Of the Fibonacci numbers less than 2^32 -- i.e., F(0) through F(47) -- F(10)=55 is the largest that has only one 0 in its binary representation, and of those not less than 2^32, the smallest one whose 32 least significant bits include fewer than 2 zero bits is Fibonacci(779038816), which exceeds 10^162809483.) - Jon E. Schoenfield, Sep 07 2015

			55 is 110111 in binary, thus 55 is in the sequence.

Cf. A004685, A221158.

Intersection of A030130 and A000045.

Select[Fibonacci@ Range[0, 120], Last@ DigitCount[#, 2] == 1 &] (* Michael De Vlieger, Sep 07 2015 *)

def count0(x):
    c = 0
    while x:
        c+= 1 - (x&1)
        if c>1:
            return 2
        x>>=1
    return c
prpr, prev = 0,1
TOP = 1<<12
print(0, end=',')
for i in range(1,TOP):
    if count0(prpr)==1:
        print(prpr, end=',')
    if (i&4095)==0:
        print('.', end=',')
    prpr, prev = prev, prpr+prev

A255568 Numbers in whose binary representation there are six 1-bits more than there are nonleading 0-bits.

Original entry on oeis.org

63, 191, 223, 239, 247, 251, 253, 254, 639, 703, 735, 751, 759, 763, 765, 766, 831, 863, 879, 887, 891, 893, 894, 927, 943, 951, 955, 957, 958, 975, 983, 987, 989, 990, 999, 1003, 1005, 1006, 1011, 1013, 1014, 1017, 1018, 1020, 2303, 2431, 2495, 2527, 2543, 2551, 2555

Offset: 1

Antti Karttunen, Mar 11 2015

Numbers for which A037861(n) = -6.

Numbers in whose binary representation (A007088) the number of 1-bits = 6 + number of (nonleading) 0 bits.

			63 ("111111" in binary) is included because there are 0 zero-bits and six 1-bits.
191 ("10111111" in binary) is included because there is 1 zero-bit and seven 1-bits, thus there are six 1-bits more than the number of 0-bits.

Antti Karttunen, Table of n, a(n) for n = 1..16303

Cf. A007088, A037861.

The intersection of A030130 and A023689 is a finite subsequence of this sequence.

is(n)=2*hammingweight(n)==6+#binary(n) \\ Charles R Greathouse IV, Dec 16 2015

for(b=3,6, for(n=2^(2*b-1),4^b-1, if(hammingweight(n)==3+b, print1(n", ")))) \\ Charles R Greathouse IV, Dec 16 2015

listBBitMembers(b)=if(b%2,return([])); my(u=List()); forvec(v=vector(3+b/2,i,[0,b-1]), listput(u, sum(i=1,#v,2^v[i])),2); Vec(u) \\ Charles R Greathouse IV, Dec 16 2015

use ntheory ":all"; my $bits = 0; for (1..1000) { my $o = hammingweight($); $bits++ if ($ & ($-1))==0; say if $o == 6+$bits-$o; } # _Dana Jacobsen, Dec 16 2015

cp's OEIS Frontend

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A086224 a(n) = 7*2^n - 1.

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A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

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A095078 Primes with a single 0 bit in their binary expansion.

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