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A354680 Terms of A354169 that are not powers of 2, in order of appearance.

%I A354680 #53 Nov 29 2023 13:08:01
%S A354680 0,3,12,17,34,68,136,768,1025,18,2080,12288,16388,72,32896,196608,
%T A354680 262400,524800,1048577,2098176,4194306,48,8390656,50331648,67112960,
%U A354680 134225920,268435460,536887296,1073741832,192,2147516416,12884901888,17179934720,34359869440
%N A354680 Terms of A354169 that are not powers of 2, in order of appearance.
%C A354680 Apart from the initial 0, all terms have Hamming weight 2. See De Vlieger et al. (2022). - _N. J. A. Sloane_, Aug 29 2022
%H A354680 Rémy Sigrist, <a href="/A354680/b354680.txt">Table of n, a(n) for n = 1..3320</a>
%H A354680 Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, <a href="http://arxiv.org/abs/2209.04108">The Binary Two-Up Sequence</a>, arXiv:2209.04108 [math.CO], Sep 11 2022.
%H A354680 Rémy Sigrist, <a href="/A354680/a354680.gp.txt">PARI program</a>
%H A354680 Rémy Sigrist, <a href="/A354680/a354680_1.gp.txt">PARI program (optimized version)</a>
%F A354680 A000120(A354169(a(n))) <> 1.
%e A354680 The initial terms of A354169 are:
%e A354680   0, 1, 2, 4, 8, 3, 16, 32, 64, 12, 128, 256.
%e A354680 The initial terms of this sequence are therefore: 0,             3,             12.
%e A354680 and the initial terms of A354798 are
%e A354680   0,             5,              9.
%o A354680 (PARI) See Links section.
%o A354680 (Python 3.10+)
%o A354680 from itertools import count, islice
%o A354680 from collections import deque
%o A354680 from functools import reduce
%o A354680 from operator import or_
%o A354680 def A354680_gen(): # generator of terms
%o A354680     aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
%o A354680     yield 0
%o A354680     while True:
%o A354680         for k in count(1):
%o A354680             m, j, j2, r, s = 0, 0, 1, b, k
%o A354680             while r > 0:
%o A354680                 r, q = divmod(r,2)
%o A354680                 if not q:
%o A354680                     s, y = divmod(s,2)
%o A354680                     m += y*j2
%o A354680                 j += 1
%o A354680                 j2 *= 2
%o A354680             if s > 0:
%o A354680                 m += s*2**b.bit_length()
%o A354680             if m not in aset:
%o A354680                 if m.bit_count() > 1:
%o A354680                     yield m
%o A354680                 aset.add(m)
%o A354680                 aqueue.append(m)
%o A354680                 if f: aqueue.popleft()
%o A354680                 b = reduce(or_,aqueue)
%o A354680                 f = not f
%o A354680                 break
%o A354680 A354680_list = list(islice(A354680_gen(),40)) # _Chai Wah Wu_, Jun 06 2022
%Y A354680 Cf. A000120, A057716, A354169, A354798 (corresponding indices).
%Y A354680 See also A354767.
%K A354680 nonn,base
%O A354680 1,2
%A A354680 _Rémy Sigrist_ and _N. J. A. Sloane_, Jun 06 2022