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

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

Offset

1,1

Comments

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.

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

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A318569

Prog

(PARI) See Links section.
(Python)
def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
def a(n):
    kn = n
    while BCR(kn) != kn: kn += n
    return kn//n
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Mar 20 2022

Crossrefs

Cf. A035928, A342582.

Sequence in context: A080079 A341707 A347820 * A336280 A362806 A277749

Adjacent sequences: A318566 A318567 A318568 * A318570 A318571 A318572

Keyword

nonn,look,base

Author

Jeffrey Shallit, Oct 12 2018

Status

approved