A053738 If k is in sequence then 2*k and 2*k+1 are not (and 1 is in the sequence); numbers with an odd number of digits in binary.

1, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 64, 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, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

Offset

1,2

Comments

Runs of successive numbers have lengths which are powers of 4.

Apparently, for any m>=1, 2^m is the largest power of 2 dividing sum(k=1,n,binomial(2k,k)^m) if and only if n is in the sequence. - Benoit Cloitre, Apr 27 2003

Numbers that begin with a 1 in base 4. - Michel Marcus, Dec 05 2013

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

Links

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

Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatsh. Math., Vol. 161, No. 1 (2010), pp. 85-114; alternative link. Section 4.3.

Formula

G.f.: x/(1-x)^2 + Sum_{k>=1} 2^(2k-1)*x^((4^k+2)/3)/(1-x). - Robert Israel, Dec 28 2016

Maple

seq(seq(i, i=4^k..2*4^k-1), k=0..5); # Robert Israel, Dec 28 2016

Mathematica

Select[Range[110], OddQ[IntegerLength[#, 2]]&] (* Harvey P. Dale, Sep 29 2012 *)

Prog

(PARI) isok(n, b=4) = digits(n, b)[1] == 1; \\ Michel Marcus, Dec 05 2013
(PARI) a(n) = n + 1<<bitor(logint(3*n, 2), 1)\3; \\ Kevin Ryde, Mar 27 2021

Crossrefs

Complement of A053754.

Cf. A029837, A079112.

Sequence in context: A294228 A342575 A046300 * A327101 A327082 A154787

Adjacent sequences: A053735 A053736 A053737 * A053739 A053740 A053741

Keyword

base,easy,nonn

Author

Henry Bottomley, Apr 06 2000

Status

approved