A352790 -id:A352790 - cp's OEIS frontend

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

Scott R. Shannon, Apr 06 2022

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.

			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.

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.

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

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 *)

A352774 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that does not share a factor with a(n-2) + a(n-1) or a(n-2)*a(n-1).

Original entry on oeis.org

1, 2, 5, 3, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 9, 35, 41, 27, 43, 47, 49, 53, 55, 59, 61, 67, 15, 71, 73, 65, 77, 51, 79, 83, 85, 89, 91, 97, 33, 101, 95, 39, 103, 107, 109, 113, 115, 119, 121, 127, 21, 125, 131, 57, 137, 139, 133, 45, 143, 149, 63, 145, 151, 69, 157, 155, 161, 81, 163, 167

Offset: 1

Scott R. Shannon, Apr 02 2022

nonn

As a(2) is even, which forces a(3) and a(4) to be odd, all following terms will be odd as the sum of two odd terms is even. Beyond a(5) = 7 all subsequent primes appear in their natural order.

			a(2) = 5 as a(1) + a(2) = 3, a(1)*a(2) = 2, and 5 is the smallest unused number that does not share a factor with 3 or 2.

Scott R. Shannon, Image of the first 5000 terms. The green line is y = n.

Cf. A352768, A352790, A332301, A351001, A349493, A251622.

A353006 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))*|a(n-1) - a(n-2)|.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 12, 14, 13, 15, 16, 31, 18, 21, 24, 20, 11, 27, 22, 25, 30, 33, 28, 35, 36, 71, 40, 37, 39, 26, 45, 38, 42, 32, 34, 44, 46, 48, 47, 50, 51, 101, 52, 17, 23, 54, 49, 55, 56, 57, 113, 58, 60, 59, 63, 61, 62, 41, 66, 65, 131, 64, 67, 69, 68, 137, 70, 72, 74

Offset: 1

Scott R. Shannon, Apr 16 2022

nonn

The sequences is conjectured to be a permutation of the positive integers. In the first 500000 terms there are 637 fixed points: 1, 2, 3, 6, 9, ..., 180611, 189383, 298097. As the terms slowly move away from the line a(n) = n as n increases it is likely no more exist.

			a(4) = 5 as (a(3)+a(2))*|a(3)-a(2)| = (3+2)*|3-2| = 5, and 5 is the smallest unused number that shares a factor with 5.
a(5) = 4 as (a(4)+a(3))*|a(4)-a(3)| = (5+3)*|5-3| = 16, and 4 is the smallest unused number that shares a factor with 16.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n - 1..2^14, showing records in red, local minima in blue, fixed points highlighted in amber, and primes in green.
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A337136, A352867, A352790, A350927.

nn = 74, c[] = 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; u = 3; Do[m = (#1 + #2) Abs[(#1 - #2)] & @@ {a[i - 1], a[i - 2]}; k = u; While[Nand[c[k] == 0, ! CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 17 2022 *)

A353082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 8, 7, 15, 11, 26, 37, 21, 16, 185, 39, 14, 25, 33, 29, 31, 6, 925, 19, 12, 217, 41, 22, 57, 35, 23, 24, 47, 71, 48, 17, 65, 32, 97, 43, 10, 53, 63, 20, 83, 103, 30, 73, 4429, 18, 401, 383, 28, 137, 55, 64, 51, 115, 128, 27, 101, 74, 45, 119, 82, 67, 149, 36, 113, 77

Offset: 1

Scott R. Shannon, Apr 22 2022

The sequence is finite, the final term being a(160)=143. See the examples. The maximum value is a(110)=35941 while the fixed points are 1, 2, 3, 8, 11, 67.

			a(6) = 9 as (a(5)+a(4))*|a(5)-a(4)| = (4+5)*|4-5| = 9, and 9 is the smallest unused number that is a divisor of 9. This is the first term that differs from A353006.
a(159) = 144, a(160) = 143. Therefore a(161) would be the smallest unused number that divides (143+144)*|143-144| = 287. The divisors are 1, 7, 41, 287, all of which have already appeared, so the sequence terminates.

Cf. A353006, A337136, A352867, A352790, A350927.

cp's OEIS Frontend

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).

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A352774 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that does not share a factor with a(n-2) + a(n-1) or a(n-2)*a(n-1).

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A353006 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))*|a(n-1) - a(n-2)|.

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A353082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

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