A349492 a(1)=1, a(2)=6; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1)+n, a(n)) > 1, gcd(a(n-1), a(n)) > 1, and gcd(n, a(n)) > 1.

1, 6, 3, 42, 470, 2, 84, 4, 78, 8, 418, 10, 598, 12, 9, 30, 1598, 14, 114, 16, 222, 18, 1886, 20, 5, 310, 2022, 22, 174, 15, 186, 24, 21, 60, 25, 610, 47878, 26, 13, 1378, 246, 27, 258, 28, 438, 32, 7426, 34, 1162, 36, 33, 90, 1166, 38, 66, 40, 582, 44, 12154, 46, 13054, 48, 39, 618, 6830

Offset: 1

Author

Scott R. Shannon, Nov 19 2021

Keywords

• nonn

Comments

The majority of terms lie near a line of gradient 1, but some terms show extremely large jumps in value, e.g., in the first 30000 terms the largest value is a(25391) = 87893333254. It is likely all numbers eventually appear although this is unknown. In the same range, and other than a(3) = 3, there are twenty-one fixed points all between 626 to 856 inclusive. As the terms for n>30000 appear both above and below the line y = n it is possible more exist although this is unknown.

Examples

			a(3) = 3 as a(2)+3 = 9, gcd(9,3)>1, gcd(6,3)>1, gcd(3,3)>1 and 3 has not been used.
a(4) = 42 as a(3)+4 = 7, gcd(7,42)>1, gcd(3,42)>1, gcd(4,42)>1 and 42 has not been used.
a(5) = 470 as a(4)+5 = 47, gcd(47,470)>1, gcd(42,470)>1, gcd(5,470)>1 and 470 has not been used.
a(6) = 2 as a(5)+6 = 476, gcd(476,2)>1, gcd(470,2)>1, gcd(6,2)>1 and 2 has not been used.
		

Links

• Scott R. Shannon, Image for the first 10000 points with values below 20000. The green line is y = n.
• Scott R. Shannon, Image of the first 30000 terms. On this scale the vast majority of terms lie along the x-axis.

Crossrefs

Cf. A119018, A064413, A349472.

Programs

Mathematica

a[1]=1; a[2]=6; a[n_]:=a[n]=(k=2;While[MemberQ[Array[a,n-1],k]||GCD[a[n-1]+n,k]<=1||GCD[a[n-1],k]<=1||GCD[n,k]<=1,k++];k); Array[a,65] (* Giorgos Kalogeropoulos, Nov 20 2021 *)

Python

from math import gcd
terms, appears = [1, 6], {6:True}
for n in range(3, 100):
    t = 2
    while not (appears.get(t) is None and gcd(terms[-1]+n, t)>1 and gcd(terms[-1], t)>1 and gcd(n, t)>1):
        t += 1
    appears[t] = True; terms.append(t)
print(terms) # Gleb Ivanov, Nov 20 2021