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.

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, 10, 177, 17, 20, 37, 19, 21, 22, 215, 39, 28, 67, 45, 26, 71, 97, 24, 77, 101, 30, 131, 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

Offset: 1

Author

Scott R. Shannon, Nov 20 2021

Keywords

• nonn

Comments

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.

Examples

			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.
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.
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.
		

Links

• Scott R. Shannon, Image of the first 100000 terms for values less than 150000. The green line is y = n.
• Scott R. Shannon, Image of the first 100000 terms.

Crossrefs

Cf. A349492, A119018, A064413, A349472.

Programs

Mathematica

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 *)

Python

from math import gcd
terms, appears = [1, 2], {2:True}
for n in range(3, 100):
    t = 3
    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):
        t += 1
    appears[t] = True; terms.append(t);
print(terms) #Gleb Ivanov, Nov 20 2021