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A353715 a(n) = b(n)+b(n+1), where b is A353709.

%I A353715 #33 Jul 07 2022 08:11:36
%S A353715 1,3,6,12,11,19,28,44,49,23,46,104,69,15,58,113,79,142,161,51,86,77,
%T A353715 43,54,92,107,167,156,90,102,61,155,226,109,157,242,354,277,63,234,
%U A353715 449,279,126,233,387,286,125,481,410,63,357,456,143,87,240,171,95,372,419,207,348,433,231,334,313,183,462,840,531,63,492,961,543,254,992,783,127
%N A353715 a(n) = b(n)+b(n+1), where b is A353709.
%C A353715 Created in an attempt to show that every number appears in A353709. For example, if one could show that the present sequence had a subsequence which was divisible by ever-increasing powers of 2, the desired result would follow.  See A353724, A353725, A353726, A353727 for more about this topic.
%H A353715 N. J. A. Sloane, <a href="/A353715/b353715.txt">Table of n, a(n) for n = 0..16383</a>
%H A353715 Walter Trump, <a href="/A353715/a353715.png">Log-log plot of first 2^24 terms.</a> Green dots (which overwrite the black dots) indicate terms a(n) which are divisible by 64.
%p A353715 g:= proc() false end: t:= 2:
%p A353715 b:= proc(n) option remember; global t; local k; if n<2 then n
%p A353715       else for k from t while g(k) or Bits[And](k, b(n-2))>0
%p A353715       or Bits[And](k, b(n-1))>0 do od; g(k):=true;
%p A353715       while g(t) do t:=t+1 od; k fi
%p A353715     end:
%p A353715 a:= n-> b(n)+b(n+1):
%p A353715 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 09 2022
%t A353715 g[_] = False ; t = 2;
%t A353715 b[n_] := b[n] = Module[{k}, If[n < 2, n,
%t A353715    For[k = t, g[k] || BitAnd[k, b[n-2]] > 0 ||
%t A353715    BitAnd[k, b[n-1]] > 0, k++]; g[k] = True;
%t A353715    While[g[t], t = t+1]; k]];
%t A353715 a[n_] := b[n] + b[n+1];
%t A353715 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jul 07 2022, after _Alois P. Heinz_ *)
%o A353715 (Python)
%o A353715 from itertools import count, islice
%o A353715 def A353715_gen(): # generator of terms
%o A353715     s, a, b, c, ab = {0,1}, 0, 1, 2, 1
%o A353715     yield 1
%o A353715     while True:
%o A353715         for n in count(c):
%o A353715             if not (n & ab or n in s):
%o A353715                 yield b+n
%o A353715                 a, b = b, n
%o A353715                 ab = a|b
%o A353715                 s.add(n)
%o A353715                 while c in s:
%o A353715                     c += 1
%o A353715                 break
%o A353715 A353715_list = list(islice(A353715_gen(),30)) # _Chai Wah Wu_, May 11 2022
%Y A353715 Cf. A353709, A353724, A353725, A353726, A353727.
%K A353715 base,nonn
%O A353715 0,2
%A A353715 _N. J. A. Sloane_, May 09 2022