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

1,1

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Patrick Wienhöft, Python program

Index entries for 2-automatic sequences.

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

Example

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

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 *)

Prog

(PARI) is(n)=n>>=valuation(n, 2); n+1!=1<<valuation(n+1, 2) \\ Charles R Greathouse IV, Mar 07 2013
(Haskell)
a101082 n = a101082_list !! (n-1)
a101082_list = filter ((> 0) . a049502) [0..]
-- Reinhard Zumkeller, Jun 17 2015
(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<<k)-(1<<i)<=x)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 23 2025

Crossrefs

Complement: A023758.

Cf. A043569, A062289, A049502.

Sequence in context: A325204 A163670 A187713 * A277706 A300669 A166934

Adjacent sequences: A101079 A101080 A101081 * A101083 A101084 A101085

Keyword

nonn,easy,changed

Author

Rick L. Shepherd, Nov 29 2004

Status

approved