reviewed

approved

proposed

reviewed

editing

proposed

(Python)

from math import gcd

from itertools import count, islice

def A353990_gen(): # generator of terms

yield 1

a, s, b = 1, 2, set()

while True:

for i in count(s):

if not (i == a+1 or i & a or gcd(i, a) > 1 or i in b):

yield i

a = i

b.add(i)

while s in b:

s += 1

break

A353990_list = list(islice(A353990_gen(), 30)) # Chai Wah Wu, May 24 2022

approved

editing

proposed

approved

editing

proposed

1, 4, 3, 8, 5, 2, 9, 16, 7, 24, 35, 12, 17, 6, 25, 32, 11, 20, 33, 10, 21, 34, 13, 18, 37, 26, 69, 40, 19, 36, 65, 14, 81, 38, 73, 22, 41, 64, 15, 112, 129, 28, 67, 44, 83, 128, 23, 72, 49, 66, 29, 96, 31, 160, 27, 68, 43, 80, 39, 88, 131, 48, 71, 56, 135, 104, 133, 50, 77, 130, 53, 74, 145, 42

Scott R. Shannon, <a href="/A353990/a353990.png">Image of the first 100000 terms</a>. The green line is y = n.

a(4) = 8 as a(3) = 3, and 8 has not yet appeared, is coprime to 3, is not 1 more than 3, while 8 = 100_1000_2 and 3 = 11_2 which have no 1-bits in common.

This sequence is similar to A093714 with the additional restriction that no term can have a 1-bit in common with the previous term in their binary expansions. This leads to the terms showing similar behavior to A109812. See the linked image.

In the first 100000 terms the fixed points are 1, 3, 5, 12, 21, 26, 44, 49, 227, 3488, 5890, 9067, 9310, 37625, 74702, although it is likely more exist. In the same range the lowest unseen number is 30686; the sequence is conjectured to be a permutation of the positive integers.

a(4) = 8 as a(3) = 3, and 8 has not yet appeared, is coprime to 3, is not 1 more than 3, while 8 = 100_2 and 3 = 11_2 which have no 1-bits in common.