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

Author

Henry Bottomley, Apr 06 2000

Keywords

• base
• easy
• nonn

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.

Crossrefs

Complement of A053754.

Cf. A029837, A079112.

Cf. A000302, A083420.

Programs

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 *)

PARI

isok(n, b=4) = digits(n, b)[1] == 1; \\ Michel Marcus, Dec 05 2013

PARI

a(n) = n + 1<Kevin Ryde, Mar 27 2021

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