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%I #52 Jun 08 2020 09:05:47
%S 1,4,5,8,9,12,13,64,65,68,69,72,73,76,77,128,129,132,133,136,137,140,
%T 141,192,193,196,197,200,201,204,205,1024,1025,1028,1029,1032,1033,
%U 1036,1037,1088,1089,1092,1093,1096,1097,1100,1101,1152
%N a(n) is the smallest number larger than a(n-1) whose a(i)-th binary digit is 0 for all i<n, with a(1)=1.
%C If n=Sum_{i=0..t} 2^(k_i) is the binary representation of n, then a(n)=Sum_{i=0..t} 2^(A335033(k_i)) is the binary representation of a(n).
%C The first 2^n-1 terms are the nonzero partial sums of 2^(A335033(1)), 2^(A335033(2)), ..., 2^(A335033(n)).
%C 2^n is in the sequence if and only if n isn't in the sequence. More specifically, 2^a(n) is not in the sequence, while 2^(A335033(n))=a(2^n).
%C Let F(n) be the number of terms in the sequence among 0,1,2,..,n-1. Then F(2^n)=2^(n-F(n)).
%C F(n)/n seems to be (1/n)^(1/log(n))^(1/loglog(n))^(1/logloglog(n))^... asymptotically (all logs are base 2).
%H Alon Heller, <a href="/A334992/b334992.txt">Table of n, a(n) for n = 1..65536</a>
%F a(Sum_{i=0..t} 2^(k_i)) = Sum_{i=0..t} 2^(A335033(k_i)) (when k_i are distinct nonnegative integers)
%e a(2)=4 because 4 is the smallest number > 1 whose 1st binary digit is 0.
%e a(6)=12 because 12 is the smallest number > 9 whose 1st, 4th, 5th, 8th, and 9th binary digits are all 0.
%o (Python)
%o def gen():
%o """ Generates the terms of A334992, starting with 1 """
%o A334992 = [0]
%o A335033 = [0]
%o while True:
%o new_power = 2**A335033[-1]
%o for i in range(len(A334992)):
%o A334992.append(A334992[i] + new_power)
%o yield A334992[i] + new_power
%o next_compl_elem = A335033[-1] + 1
%o while next_compl_elem in A334992:
%o next_compl_elem += 1
%o A335033.append(next_compl_elem)
%o def A334992_list(n):
%o """ Returns the n first elements as a list """
%o g = gen()
%o return [next(g) for _ in range(n)]
%Y Complement of A335033.
%K nonn,base,easy
%O 1,2
%A _Alon Heller_, May 20 2020
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