A160502 -id:A160502 - cp's OEIS frontend

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

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

A030130 Binary expansion contains a single 0.

Original entry on oeis.org

0, 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

Offset: 1

Toby Donaldson (tjdonald(AT)uwaterloo.ca)

From Reinhard Zumkeller, Aug 29 2009: (Start)
A023416(a(n)) = 1;
apart from the initial term the sequence can be seen as a triangle read by rows, see A164874;
A055010 and A086224 are subsequences, see also A000918 and A036563. (End)
Zero and numbers of form 2^m-2^k-1, 2 <= m, 0 <= k <= m-2. - Zak Seidov, Aug 06 2010

			23 is OK because it is '10111' in base 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000918, A023416, A036563, A055010, A086224, A140977, A160502, A164874, A190620.

long int element (long int i) { return (pow(2,g(i))-1-pow(2,(pow(2*g(i)-1,2)-1-8*i)/8));} long int g(long int m) {if (m==0) return(1); return ((sqrt(8*m-7)+3)/2);}

a030130 n = a030130_list !! (n-1)
a030130_list = filter ((== 1) . a023416) [0..]
-- Reinhard Zumkeller, Mar 31 2015, Dec 07 2012

[0] cat [k:k in [0..2050]| Multiplicity(Intseq(k,2),0) eq 1]; // Marius A. Burtea, Feb 06 2020

Sort[Flatten[{{0}, Table[2^n - 2^m - 1, {n, 2, 50}, {m, 0, n - 2}]}]] (* Zak Seidov, Aug 06 2010 *)
Select[Range[0,2100],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Dec 19 2021 *)

print1("0, ");for(k=1,2039,my(v=digits(k,2));if(vecsum(v)==#v-1,print1(k,", "))) \\ Hugo Pfoertner, Feb 06 2020

from math import isqrt, comb
def A030130(n): return (1<<(a:=(isqrt(n-1<<3)+1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

a(n) = 2^(g(n))-1-2^(((2*g(n)-1)^2-1-8*n)/8) with g(n)=int((sqrt(8*n-7)+3)/2) for all n>0 and g(0)=1. - Ulrich Schimke (ulrschimke(AT)aol.com)
a(n+1) = A140977(a(n)) for any n > 1. - Rémy Sigrist, Feb 06 2020
Sum_{n>=2} 1/a(n) = A160502. - Amiram Eldar, Oct 06 2020
a(n) = (A190620(n-1)-1)/2. - Chai Wah Wu, Dec 19 2024

More terms from Erich Friedman

Offset fixed by Reinhard Zumkeller, Aug 24 2009

A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2010

Obviously for k > 0 in base 2 having no bit equal to 1 the sum is 0 and for 1 bit equal to 1 the sum is 2.

			Sum_{k>0} 1/A018900(k) = 1.52899956069688841838263949451...

Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Cf. A065442, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839, A140502, A160502, A018900, A323482.

 evalf( 2*add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022

(* first install irwinSums.m, see either reference, then *) First@ RealDigits@ iSum[1, 2, 2^7, 2]

Equals Sum_{j>=1} Sum_{i=0..j-1} 1/(2^i + 2^j).
From Amiram Eldar, Jun 30 2020: (Start)
Equals Sum_{k>=0} 1/(2^k + 1/2).
Equals 2 * A323482 - 1. (End)
Equals 2*Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

A179954 Decimal expansion of the sum of the reciprocals of pandigital numbers in which each digit appears exactly once.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2010

This is example in 3. 1(a) of R. Baillie, revised.

This is a finite sum so it is a rational number.

			0.0008258903479192529386079577501789135432537929965887385729771528345968177...

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series
Eric Weisstein's World of Mathematics, Digit
Wikipedia, Kempner series

Cf. A050278, A010784, A082830-A082839, A140502, A160502.

Sum_{k=1..3265920} 1/A050278(k).

Standardized offset and leading zeros from R. J. Mathar, Aug 06 2010

More terms from Robert G. Wilson v, Sep 07 2010

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

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A030130 Binary expansion contains a single 0.

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A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

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A179954 Decimal expansion of the sum of the reciprocals of pandigital numbers in which each digit appears exactly once.

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