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a(n) is the smallest positive integer such that n*a(n) is a "binary antipalindrome" (i.e., an element of A035928).
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%I #51 Mar 20 2022 18:29:21

%S 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 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 58,264,1,850,4,162,4,16,9,34,63,38,127,56,3,42,39,2

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

%C 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 Then n divides (2^{k*phi(r)} - 1)*2^{k*phi(r)}, which is a binary antipalindrome.

%H Rémy Sigrist, <a href="/A318569/b318569.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A318569/a318569.gp.txt">PARI program for A318569</a>

%o (PARI) See Links section.

%o (Python)

%o def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})

%o def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)

%o def a(n):

%o kn = n

%o while BCR(kn) != kn: kn += n

%o return kn//n

%o print([a(n) for n in range(1, 72)]) # _Michael S. Branicky_, Mar 20 2022

%Y Cf. A035928, A342582.

%K nonn,look,base

%O 1,1

%A _Jeffrey Shallit_, Oct 12 2018