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

Author

Jeffrey Shallit, Oct 12 2018

Keywords

• nonn
• look
• base

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

Crossrefs

Cf. A035928, A342582.

Programs

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