A087116 -id:A087116 - cp's OEIS frontend

A173589 Integers whose binary representation contains exactly three 1's, no two 1's being adjacent.

21, 37, 41, 42, 69, 73, 74, 81, 82, 84, 133, 137, 138, 145, 146, 148, 161, 162, 164, 168, 261, 265, 266, 273, 274, 276, 289, 290, 292, 296, 321, 322, 324, 328, 336, 517, 521, 522, 529, 530, 532, 545, 546, 548, 552, 577, 578, 580, 584, 592, 641, 642, 644, 648

Offset: 1

David Koslicki (koslicki(AT)math.psu.edu), Feb 22 2010

Subsequence of A014311. [R. J. Mathar, Feb 24 2010]

A000120(a(n))=3; A023416(a(n))>1; 1 < A087116(a(n))<=3. [Reinhard Zumkeller, Mar 11 2010]

			a(1) = 21 = 10101_2.
a(2) = 37 = 100101_2.
a(3) = 41 = 101001_2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A014311, A023416, A087116.

seq(seq(seq(2^a+2^b+2^c, c=0..b-2),b=2..a-2),a=4..10); # Robert Israel, Dec 19 2016

e31sQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Total[idn2]==3 && SequenceCount[ idn2,{1,1}]==0]; Select[Range[700],e31sQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 20 2018 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A173589(n): return (1<<(r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1)+1)+(1<Chai Wah Wu, Apr 07 2025

More terms from R. J. Mathar, Feb 24 2010

A087120 Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.

Original entry on oeis.org

1, 0, 10, 42, 170, 682, 2730, 10922, 43690, 174762, 699050, 2796202, 11184810, 44739242, 178956970, 715827882, 2863311530, 11453246122, 45812984490, 183251937962, 733007751850, 2932031007402, 11728124029610, 46912496118442

Offset: 0

Reinhard Zumkeller, Aug 14 2003

nonn

A087116(a(n))=n and A087116(k)

For n>1, a(n) = A020988(n-1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A087118, A023416, A007088.

Join[{1,0},NestList[4#+2&,10,25]] (* Harvey P. Dale, Apr 20 2019 *)

a(0)=1, a(1)=0, a(2)=10, a(n)=4*a(n-1)+2.

A087119 Numbers having more than one maximal group of consecutive zeros in binary representation of n.

Original entry on oeis.org

10, 18, 20, 21, 22, 26, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 50, 52, 53, 54, 58, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 98, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 114, 116, 117, 118, 122

Offset: 1

Reinhard Zumkeller, Aug 14 2003

nonn

A087116(a(n)) > 1.

Index entries for sequences related to binary expansion of n

Cf. A087118, A023416, A007088.

A306874 Lexicographically earliest sequence of distinct positive terms such that the binary representation of the bitwise-OR of two consecutive terms has exactly one run of consecutive zeros.

Original entry on oeis.org

1, 4, 2, 6, 8, 3, 9, 5, 12, 10, 11, 16, 7, 17, 13, 20, 14, 18, 19, 21, 22, 23, 32, 15, 33, 24, 25, 26, 27, 34, 28, 29, 36, 30, 38, 35, 37, 39, 40, 47, 41, 46, 43, 44, 48, 45, 42, 49, 50, 51, 52, 55, 53, 54, 56, 57, 58, 59, 64, 31, 65, 60, 61, 68, 62, 66, 67

Offset: 1

Rémy Sigrist, Mar 14 2019

This sequence is a variant of A306869.

			The first terms, alongside the binary representation of a(n) OR a(n+1), are:
  n   a(n)  bin(a(n) OR a(n+1))
  --  ----  -------------------
   1     1                101
   2     4                110
   3     2                110
   4     6               1110
   5     8               1011
   6     3               1011
   7     9               1101
   8     5               1101
   9    12               1110
  10    10               1011
  11    11              11011
  12    16              10111
  13     7              10111
  14    17              11101
  15    13              11101

Rémy Sigrist, Table of n, a(n) for n = 1..16384
Rémy Sigrist, PARI program for A306874

Cf. A087116, A306869.

PARI
```
See Links section.
```

A087116(a(n) OR a(n+1)) = 1.

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A173589 Integers whose binary representation contains exactly three 1's, no two 1's being adjacent.

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A087120 Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.

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A087119 Numbers having more than one maximal group of consecutive zeros in binary representation of n.

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A306874 Lexicographically earliest sequence of distinct positive terms such that the binary representation of the bitwise-OR of two consecutive terms has exactly one run of consecutive zeros.

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