cp's OEIS Frontend

A353990 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1), does not equal a(n-1)+1, and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

%I A353990 #14 May 24 2022 09:12:45
%S A353990 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,
%T A353990 69,40,19,36,65,14,81,38,73,22,41,64,15,112,129,28,67,44,83,128,23,72,
%U A353990 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
%N A353990 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1), does not equal a(n-1)+1, and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).
%C A353990 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.
%C A353990 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.
%H A353990 Scott R. Shannon, <a href="/A353990/a353990.png">Image of the first 100000 terms</a>. The green line is y = n.
%e A353990 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 = 1000_2 and 3 = 11_2 which have no 1-bits in common.
%o A353990 (Python)
%o A353990 from math import gcd
%o A353990 from itertools import count, islice
%o A353990 def A353990_gen(): # generator of terms
%o A353990     yield 1
%o A353990     a, s, b = 1, 2, set()
%o A353990     while True:
%o A353990         for i in count(s):
%o A353990             if not (i == a+1 or i & a or gcd(i,a) > 1 or i in b):
%o A353990                 yield i
%o A353990                 a = i
%o A353990                 b.add(i)
%o A353990                 while s in b:
%o A353990                     s += 1
%o A353990                 break
%o A353990 A353990_list = list(islice(A353990_gen(),30)) # _Chai Wah Wu_, May 24 2022
%Y A353990 Cf. A093714, A109812, A353989, A352763.
%K A353990 nonn,base
%O A353990 1,2
%A A353990 _Scott R. Shannon_, May 13 2022