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%I #22 Mar 24 2023 23:12:29
%S 0,2,3,6,7,10,11,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,
%T 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U 54,55,56,57,58,59,60,61,62,63,66,67
%N a(n) is the smallest number larger than a(n-1) that is not a partial sum of 2^a(1),2^a(2),...,2^a(n-1), with a(1)=0.
%C a(n) is the smallest number such that a(n)>a(n-1) and a(n)'s k-th binary digit is 1, for some k different from a(1),a(2),...,a(n-1).
%C The first 2^n-1 terms of A334992 are the nonzero partial sums of {2^a(1), 2^a(2), ..., 2^a(n)}.
%C 2^n is in the sequence if and only if n isn't in the sequence.
%C Let G(n) be the number of terms in the sequence among 0,1,2,..,n-1. Then G(2^n)=2^n-2^G(n).
%C G(n)/n seems to be 1-(1/n)^(1/log(n))^(1/loglog(n))^(1/logloglog(n))^... asymptotically (all logs are base 2).
%H Alon Heller, <a href="/A335033/b335033.txt">Table of n, a(n) for n = 1..65536</a>
%e a(2)=2 because 2 is the smallest number > 0 which isn't a partial sum of {1}={2^0}.
%e a(4)=6 because 6 is the smallest number > 3 which isn't a partial sum of {1,4,8}={2^0,2^2,2^3}.
%o (Python)
%o def gen():
%o """Generates the terms of A335033, starting with 1"""
%o A334992 = [0, 1]
%o A335033 = [0, 2]
%o yield 0
%o yield 2
%o power_index = 1
%o while True:
%o new_power = 2 ** A335033[power_index]
%o for i in range(len(A334992)):
%o A334992.append(A334992[i] + new_power)
%o for x in range(A334992[-2] + 1, A334992[-1]):
%o if A335033[-1] != x:
%o A335033.append(x)
%o yield x
%o power_index += 1
%o def A335033_list(n):
%o """Returns the n first elements as a list"""
%o g = gen()
%o return [next(g) for _ in range(n)]
%o print(A335033_list(20))
%Y Complement of A334992.
%K nonn,base,easy
%O 1,2
%A _Alon Heller_, May 20 2020