A101082 Numbers n such that binary representation contains bit strings "10" and "01" (possibly overlapping).

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Author

Rick L. Shepherd, Nov 29 2004

Keywords

• nonn
• easy

Comments

Subsequence of A062289; set difference A062289 minus A043569.

Complement of A023758. Also numbers not the sum of consecutive powers of 2. - Omar E. Pol, Mar 04 2013

Equivalently, numbers not the difference of two powers of two. - Charles R Greathouse IV, Mar 07 2013

The terms >=9 are bases in which a power of 2 exists, which does not contain a digit that is a power of 2. In base 10, 2^16 = 65536 is such a number, as it does not contain any one-digit power of 2, which in base 10 are 1, 2, 4 and 8. - Patrick Wienhöft, Jul 28 2016

Examples

			29 = 11101_2 is a term, "10" and "01" are contained (here overlapping).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Patrick Wienhöft, Python program
• Index entries for 2-automatic sequences.

Crossrefs

Complement: A023758.

Cf. A043569, A062289, A049502.

Programs

Haskell

a101082 n = a101082_list !! (n-1)
a101082_list = filter ((> 0) . a049502) [0..]
-- Reinhard Zumkeller, Jun 17 2015

Mathematica

Select[Range@ 120, Function[d, Times @@ Total@ Map[Map[Function[k, Boole@ MatchQ[#, k]], {{1, 0}, {0, 1}}] &, Partition[d, 2, 1]] > 0]@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Dec 23 2016 *)
Select[Range[100],With[{c=IntegerDigits[#,2]},SequenceCount[c,{1,0}]>0&&SequenceCount[c,{0,1}]>0]&] (* Harvey P. Dale, Jun 08 2025 *)

PARI

is(n)=n>>=valuation(n,2);n+1!=1<Charles R Greathouse IV, Mar 07 2013

Python

def A101082(n):
    def f(x): return n+((k:=x.bit_length())*(k-1)>>1)+sum(1 for i in range(k) if (1<Chai Wah Wu, Feb 23 2025

Formula

a(n) ~ n. In particular a(n) = n + (log_2 n)^2/2 + O(log n). - Charles R Greathouse IV, Mar 07 2013

A049502(a(n)) > 0. - Reinhard Zumkeller, Jun 17 2015