A358444 a(1) = 1, a(2) = 2; for n > 2, a(n) = smallest positive number which has not appeared that has a common factor with a(n-2)^2 + a(n-1)^2.

1, 2, 5, 29, 4, 857, 10, 734549, 539562233501, 6, 12433, 15, 8, 17, 353, 12, 124753, 13, 14, 20, 16, 18, 22, 24, 25, 1201, 26, 41, 2357, 28, 5556233, 37, 30, 2269, 39, 32, 35, 52, 3929, 40, 15438641, 82, 45, 65, 34, 5381, 78, 50, 36, 38, 42, 44, 46, 48, 51, 3, 9, 21, 27, 33, 54, 55, 91

Offset: 1

Author

Scott R. Shannon, Nov 16 2022

Keywords

• nonn

Comments

The majority of terms are concentrated along or just above the line a(n) = n, resulting in 51 fixed points in the first 5000 terms. However, some terms are much larger because the sum of the squares of the previous two terms is a prime number.

Conjecture: the sequence is a permutation of the positive integers.

Examples

			a(5) = 4 as a(3)^2 + a(4)^2 = 25 + 841 = 866, and 4 is the smallest unused number that shares a factor with 866.
a(9) = 539562233501 as a(7)^2 + a(8)^2 = 100 + 539562233401 = 539562233501, which is a prime number.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 10000 terms where a(n) is within 10% of n. The green line is a(n) = n.
• Michael De Vlieger, Log-log scatterplot of a(n), n = 1..24857. a(24858) is a multiple of a prime factor of 2345424289569907866042152579118178340801^2 + 24922^2.
• Michael De Vlieger, Table of n, a(n) for n = 1..65536.

Crossrefs

Cf. A337136, A347594, A098550, A336957.

Programs

Mathematica

nn = 120; c[] = False; q[] = 1; Do[Set[{a[i], c[i], q[i]}, {i, True, 2}], {i, 2}]; i = a[1]^2; j = a[2]^2; Do[k = i + j; s = FactorInteger[k][[All, 1]]; Do[(m = q[#]; While[c[# m], m++]; q[#] = m; If[# m < k, k = # m]) &[s[[n]]], {n, Length[s]}]; Set[{a[n], c[k], i, j}, {k, True, j, k^2}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 17 2022 *)