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A318569 a(n) is the smallest positive integer such that n*a(n) is a "binary antipalindrome" (i.e., an element of A035928).

%I A318569 #51 Mar 20 2022 18:29:21
%S A318569 2,1,4,3,2,2,6,7,62,1,62,1,4,3,10,15,10,31,2,12,2,31,26,10,6,2,116,2,
%T A318569 8,5,18,31,254,5,78,26,18,1,24,6,18,1,70,86,11894,13,254,5,46,3,4,1,4,
%U A318569 58,264,1,850,4,162,4,16,9,34,63,38,127,56,3,42,39,2
%N A318569 a(n) is the smallest positive integer such that n*a(n) is a "binary antipalindrome" (i.e., an element of A035928).
%C A318569 a(n) exists:  write n = r*2^i, where r is odd.  Then r divides 2^phi(r) - 1, where phi is the Euler phi function.  Choose k such that k phi(r) >= i.
%C A318569 Then n divides (2^{k*phi(r)} - 1)*2^{k*phi(r)}, which is a binary antipalindrome.
%H A318569 Rémy Sigrist, <a href="/A318569/b318569.txt">Table of n, a(n) for n = 1..10000</a>
%H A318569 Rémy Sigrist, <a href="/A318569/a318569.gp.txt">PARI program for A318569</a>
%o A318569 (PARI) See Links section.
%o A318569 (Python)
%o A318569 def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
%o A318569 def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
%o A318569 def a(n):
%o A318569     kn = n
%o A318569     while BCR(kn) != kn: kn += n
%o A318569     return kn//n
%o A318569 print([a(n) for n in range(1, 72)]) # _Michael S. Branicky_, Mar 20 2022
%Y A318569 Cf. A035928, A342582.
%K A318569 nonn,look,base
%O A318569 1,1
%A A318569 _Jeffrey Shallit_, Oct 12 2018