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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.

%I A053738 #38 Feb 16 2025 16:10:41
%S A053738 1,4,5,6,7,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,64,65,66,
%T A053738 67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,
%U A053738 90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109
%N 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.
%C A053738 Runs of successive numbers have lengths which are powers of 4.
%C A053738 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
%C A053738 Numbers that begin with a 1 in base 4. - _Michel Marcus_, Dec 05 2013
%C A053738 The lower and upper asymptotic densities of this sequence are 1/3 and 2/3, respectively. - _Amiram Eldar_, Feb 01 2021
%H A053738 Robert Israel, <a href="/A053738/b053738.txt">Table of n, a(n) for n = 1..10000</a>
%H A053738 Manfred Madritsch and Stephan Wagner, <a href="https://doi.org/10.1007/s00605-009-0126-y">A central limit theorem for integer partitions</a>, Monatsh. Math., Vol. 161, No. 1 (2010), pp. 85-114; <a href="https://www.researchgate.net/publication/225845584_A_central_limit_theorem_for_integer_partitions">alternative link</a>. Section 4.3.
%F A053738 G.f.: x/(1-x)^2 + Sum_{k>=1} 2^(2k-1)*x^((4^k+2)/3)/(1-x). - _Robert Israel_, Dec 28 2016
%p A053738 seq(seq(i,i=4^k..2*4^k-1),k=0..5); # _Robert Israel_, Dec 28 2016
%t A053738 Select[Range[110],OddQ[IntegerLength[#,2]]&] (* _Harvey P. Dale_, Sep 29 2012 *)
%o A053738 (PARI) isok(n, b=4) = digits(n, b)[1] == 1; \\ _Michel Marcus_, Dec 05 2013
%o A053738 (PARI) a(n) = n + 1<<bitor(logint(3*n,2),1)\3; \\ _Kevin Ryde_, Mar 27 2021
%Y A053738 Complement of A053754.
%Y A053738 Cf. A029837, A079112.
%Y A053738 Cf. A000302, A083420.
%K A053738 base,easy,nonn
%O A053738 1,2
%A A053738 _Henry Bottomley_, Apr 06 2000