A053754 If k is in the sequence then 2*k and 2*k+1 are not (and 0 is in the sequence); when written in binary k has an even number of bits (0 has 0 digits).

0, 2, 3, 8, 9, 10, 11, 12, 13, 14, 15, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148

Offset

1,2

Comments

Runs of successive terms with same number of bits have length twice powers of 4 (A081294). [Clarified by Michel Marcus, Oct 21 2020]

The sequence A081294 counts compositions of even numbers - Gus Wiseman, Aug 12 2021

A031443 is a subsequence; A179888 is the intersection of this sequence and A032925. - Reinhard Zumkeller, Jul 31 2010

The lower and upper asymptotic densities of this sequence are 1/3 and 2/3, respectively. - Amiram Eldar, Feb 01 2021

From Gus Wiseman, Aug 10 2021: (Start)

Also numbers k such that the k-th composition in standard order (row k of A066099) has even sum. The terms and corresponding compositions begin:

0: () 2: (2) 8: (4)

3: (1,1) 9: (3,1)

10: (2,2)

11: (2,1,1)

12: (1,3)

13: (1,2,1)

14: (1,1,2)

15: (1,1,1,1)

The following pertain to compositions in standard order: A000120, A029837, A070939, A066099, A124767.

(End)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001

Mathematica

Select[Range[0, 150], EvenQ @ IntegerLength[#, 2] &] (* Amiram Eldar, Feb 01 2021 *)

Prog

(Haskell)
a053754 n = a053754_list !! (n-1)
a053754_list = 0 : filter (even . a070939) [1..]
-- Reinhard Zumkeller, Apr 18 2015
(PARI) lista(nn) = {my(va = vector(nn)); for (n=2, nn, my(k=va[n-1]+1); while (#select(x->(x==k\2), va), k++); va[n] = k; ); va; } \\ Michel Marcus, Oct 20 2020
(PARI) a(n) = n-1 + (1<<bitand(logint(6*n-3, 2), -2))\3; \\ Kevin Ryde, Apr 30 2021

Crossrefs

Cf. A031443, A032925, A179888.

Positions of even terms in A029837 with offset 0.

The complement (the odd version) is A053738, counted by A000302.

The version for Heinz numbers of partitions is A300061, counted by A058696.

Cf. A000120, A001969, A008549, A070939, A080791, A114121.

Sequence in context: A291838 A114585 A161641 * A353652 A080745 A283776

Adjacent sequences: A053751 A053752 A053753 * A053755 A053756 A053757

Keyword

base,easy,nonn

Author

Henry Bottomley, Apr 06 2000

Extensions

Offset corrected by Reinhard Zumkeller, Jul 30 2010

Status

approved