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%I #17 Nov 20 2021 20:22:41
%S 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 10,177,17,20,37,19,21,22,215,39,28,67,45,26,71,97,24,77,101,30,131,
%U 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 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 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 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 Scott R. Shannon, <a href="/A349493/a349493_1.png">Image of the first 100000 terms</a>.
%e 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 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 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 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 (Python)
%o from math import gcd
%o terms, appears = [1, 2], {2:True}
%o for n in range(3, 100):
%o t = 3
%o 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 t += 1
%o appears[t] = True; terms.append(t);
%o print(terms) #_Gleb Ivanov_, Nov 20 2021
%Y Cf. A349492, A119018, A064413, A349472.
%K nonn
%O 1,2
%A _Scott R. Shannon_, Nov 20 2021
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