A352867 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1), a(n-2), and a(n-1)+a(n-2).

1, 2, 6, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 105, 7, 21, 35, 49, 63, 77, 91, 119, 126, 133, 140, 114, 116

Offset: 1

Author

Scott R. Shannon, Apr 06 2022

Keywords

• nonn
• look

Comments

The sequence shows long runs of even terms differing by 2 which are eventually broken by a number with a product of odd primes less than the last even term. The term after such run-breaking terms is often significantly less than the previous terms, leading to the sequence showing abrupt dips in its values. In the first 200000 terms the longest even-numbered run is 106 terms, and it is likely these runs can grow arbitrarily long. Likewise long runs of odd terms also exist, the longest such run being 133 terms in the same range. However unlike the even-numbered runs which increase by 2 each term the odd-numbered runs increase with differing amounts between each term. Between the large dips in value the majority of terms are concentrated along a line with gradient ~ 1.125. See the linked images.

It takes many terms for the primes to appear, e.g. a(166) = 3, a(239) = 5, a(1841) = 23, a(13325) = 61, a(158205) = 191. They do not appear in their natural order.

Other than the first few terms the only fixed point up to 200000 terms is 63. It is possible more exist although this is unknown. The sequence is almost certainly a permutation of the positive integers.

Examples

			a(4) = 4 as a(2)=2, a(3)=6, a(2)+a(3)=8, and 4 is the smallest unused number that shares a factor with 2, 6, and 8.
a(58) = 105 as a(56)=110, a(57)=112, a(56)+a(57)=222, and 105 = 3*5*7 is the smallest unused number that shares a factor with 110, 112, and 222. This breaks a run of fifty-three consecutive even terms differing by 2.
a(59) = 7 as a(57)=112, a(58)=105, a(57)+a(58)=217, and 7 is the smallest unused number that shares a factor with 112, 105, and 217. This is the second prime to appear after a(2)=2.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^10, showing intervals of terms with the same g = A007947(gcd(a(n-2), a(n-1), a(n-2)+a(n-1))) in a color function where g = 2 is black, g = 3 is red, g = 5 orange, etc. The first term in the intervals are noted in black, their index in italic below the point.
• Scott R. Shannon, Image of the first 1000 terms. The green line is y = n.
• Scott R. Shannon, Image of the first 200000 terms.
• Scott R. Shannon, Image of the first 200000 terms with parity. White are even terms, red are odd terms.

Crossrefs

Cf. A352768, A349493, A352790, A352774, A351001, A251622, A064413.

Programs

Mathematica

nn = 71; u = 1; c[] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 2, 6}]; While[c[u] > 0, u++]; Do[k = u; While[Nand[c[k] == 0, ! CoprimeQ[#1, k], ! CoprimeQ[#2, k], ! CoprimeQ[#3, k]], k++] & @@ {#1, #2, #1 + #2} & @@ {a[i - 2], a[i - 1]}; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 4, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 28 2022 *)