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A355166
Lexicographically earliest sequence of distinct positive integers such for any n > 0, n and a(n) are coprime and have no common 1-bits in their binary expansions.
1
2, 1, 4, 3, 8, 17, 16, 5, 20, 21, 32, 19, 18, 33, 64, 7, 6, 13, 12, 9, 10, 41, 40, 35, 34, 37, 68, 65, 66, 97, 96, 11, 14, 25, 24, 67, 26, 73, 80, 23, 22, 85, 84, 81, 82, 129, 128, 71, 72, 69, 76, 75, 74, 137, 136, 131, 70, 133, 132, 193, 130, 257, 256, 15, 28
OFFSET
1,1
COMMENTS
This sequence combines features of A065190 and of A238757.
This sequence is a self-inverse permutation of the nonnegative integers, without fixed points.
This sequence is well defined:
- if n is odd, then we can extend the sequence with a power of 2 > n,
- if n < 2^k is even, then we can extend the sequence with a prime number of the form 1 + t*2^k (Dirichlet's theorem on arithmetic progressions guarantees us that there is an infinity of such prime numbers).
When n is odd, a(n) is even and vice-versa.
EXAMPLE
The first terms, alongside binary expansions and distinct prime factors, are:
n a(n) bin(n) bin(a(n)) dpf(n) dpf(a(n))
-- ---- ------ --------- ------ ---------
1 2 1 10 {} {2}
2 1 10 1 {2} {}
3 4 11 100 {3} {2}
4 3 100 11 {2} {3}
5 8 101 1000 {5} {2}
6 17 110 10001 {2, 3} {17}
7 16 111 10000 {7} {2}
8 5 1000 101 {2} {5}
9 20 1001 10100 {3} {2, 5}
10 21 1010 10101 {2, 5} {3, 7}
PROG
(PARI) See Links section.
(Python)
from math import gcd
from itertools import count, islice
def agen(): # generator of terms
aset, mink = set(), 1
for n in count(1):
an = mink
while an in aset or n&an or gcd(n, an)!=1: an += 1
aset.add(an); yield an
while mink in aset: mink += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Jun 22 2022
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Jun 22 2022
STATUS
approved