cp's OEIS Frontend

A349493 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-2)+a(n-1), a(n)) > 1 while gcd(a(n-2), a(n)) = 1 and gcd(a(n-1), a(n)) = 1.

%I A349493 #17 Nov 20 2021 20:22:41
%S A349493 1,2,3,5,4,9,13,8,7,15,11,14,25,27,16,43,59,6,35,41,12,53,55,18,73,49,
%T A349493 10,177,17,20,37,19,21,22,215,39,28,67,45,26,71,97,24,77,101,30,131,
%U A349493 23,32,33,65,34,57,91,40,393,433,38,51,89,44,63,107,46,75,121,52,173,69,50,119,117,58,85
%N A349493 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-2)+a(n-1), a(n)) > 1 while gcd(a(n-2), a(n)) = 1 and gcd(a(n-1), a(n)) = 1.
%C A349493 In the first 100000 terms the smallest unseen number is 14657, although it is likely all numbers eventually appear. In the same range the fixed points are 3, 8, 11, 69, 207, 543, 555, 663, 687, 981. The majority of terms more than n = 100000 appear to move away from the line y = n, see the linked image, so it is unclear if more exist. The largest value in the first 100000 terms is a(87952) = 4758245.
%H A349493 Scott R. Shannon, <a href="/A349493/a349493.png">Image of the first 100000 terms for values less than 150000</a>. The green line is y = n.
%H A349493 Scott R. Shannon, <a href="/A349493/a349493_1.png">Image of the first 100000 terms</a>.
%e A349493 a(3) = 3 as a(1)+a(2) = 3, gcd(1,3) = 1, gcd(2,3) = 1, gcd(3,3) > 1 and 3 is unused.
%e A349493 a(4) = 5 as a(2)+a(3) = 5, gcd(2,5) = 1, gcd(3,5) = 1, gcd(5,5) > 1 and 5 is unused.
%e A349493 a(8) = 8 as a(6)+a(7) = 22, gcd(9,8) = 1, gcd(13,8) = 1, gcd(22,8) > 1 and 8 is unused.
%t A349493 a[1]=1; a[2]=2; a[n_]:=a[n]=(k=2;While[MemberQ[Array[a,n-1],k]||GCD[a[n-2]+a[n-1],k]<=1||GCD[a[n-2],k]!=1||GCD[a[n-1],k]!=1,k++];k); Array[a,74] (* _Giorgos Kalogeropoulos_, Nov 20 2021 *)
%o A349493 (Python)
%o A349493 from math import gcd
%o A349493 terms, appears = [1, 2], {2:True}
%o A349493 for n in range(3, 100):
%o A349493     t = 3
%o A349493     while not(appears.get(t) is None and gcd(terms[-2]+terms[-1], t)>1 and gcd(terms[-2], t)==1 and gcd(terms[-1], t)==1):
%o A349493         t += 1
%o A349493     appears[t] = True; terms.append(t);
%o A349493 print(terms) #_Gleb Ivanov_, Nov 20 2021
%Y A349493 Cf. A349492, A119018, A064413, A349472.
%K A349493 nonn
%O A349493 1,2
%A A349493 _Scott R. Shannon_, Nov 20 2021