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A360388 Positive integers with binary expansion (b(1), ..., b(m)) such that Sum_{i = 1..m-k} b(i)*b(i+k) is odd for all k = 0..m-1.

%I A360388 #26 Feb 07 2023 17:53:11
%S A360388 1,11,13,2787,3189,36783,37063,43331,47803,49813,56669,58121,62961,
%T A360388 9205487,16215601,23070091,23248907,27264653,27475981,43469906355,
%U A360388 55167946629,75985591407,80056245671,81489328999,83389490039,87235136243,88437433811,90400346819
%N A360388 Positive integers with binary expansion (b(1), ..., b(m)) such that Sum_{i = 1..m-k} b(i)*b(i+k) is odd for all k = 0..m-1.
%C A360388 Leading zeros in binary expansions are ignored.
%C A360388 All terms are odd and odious (A092246).
%C A360388 This sequence is infinite since we can, from a given term, build another larger term (see Guy reference).
%C A360388 See A053006 for the distinct binary lengths.
%C A360388 If m is a term, then A030101(m) is also a term.
%D A360388 R. K. Guy, Unsolved Problems in Number Theory, E38.
%H A360388 Rémy Sigrist, <a href="/A360388/a360388.gp.txt">PARI program</a>
%H A360388 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A360388 For n = 11:
%e A360388 - the binary expansion of 11 is b = (1,1,0,1),
%e A360388 - b(1)*b(1) + b(2)*b(2) + b(3)*b(3) + b(4)*b(4) = 1 + 1 + 0 + 1 = 3 is odd,
%e A360388 - b(1)*b(2) + b(2)*b(3) + b(3)*b(4) = 1 + 0 + 0 = 1 is odd,
%e A360388 - b(1)*b(3) + b(2)*b(4) = 0 + 1 = 1 is odd,
%e A360388 - b(1)*b(4) = 1 is odd,
%e A360388 - so 11 belongs to the sequence.
%o A360388 (PARI) See Links section.
%o A360388 (Python)
%o A360388 from itertools import count, islice
%o A360388 from functools import reduce
%o A360388 from operator import ixor
%o A360388 def A360388_gen(startvalue=1): # generator of terms >= startvalue
%o A360388     for n in count(max(startvalue,1)):
%o A360388         b = tuple(int(d) for d in bin(n)[2:])
%o A360388         m = len(b)
%o A360388         if all(reduce(ixor, (b[i]&b[i+k] for i in range(m-k))) for k in range(m)):
%o A360388             yield n
%o A360388 A360388_list = list(islice(A360388_gen(),10)) # _Chai Wah Wu_, Feb 07 2023
%Y A360388 Cf. A030101, A053006, A092246.
%K A360388 nonn,base
%O A360388 1,2
%A A360388 _Rémy Sigrist_, Feb 05 2023