A354739 -id:A354739 - cp's OEIS frontend

A354727 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

1, 2, 4, 8, 6, 3, 9, 12, 20, 5, 10, 22, 11, 33, 15, 24, 14, 7, 21, 28, 16, 26, 13, 39, 18, 34, 17, 51, 27, 42, 36, 32, 50, 25, 45, 40, 64, 30, 55, 35, 56, 48, 75, 66, 38, 19, 57, 76, 44, 72, 46, 23, 69, 92, 52, 65, 95, 60, 93, 31, 62, 94, 47, 141, 54, 90, 63, 49, 84, 68, 85, 125, 70, 112, 58

Offset: 1

Scott R. Shannon, Jun 05 2022

nonn

This sequences uses similar a similar rule to A354687 but here the sign of the difference between a(n-1) and a(n) is considered. The terms show an overall similar behavior to A354687 although here only two lines show a higher concentration of terms than the other lines. See the linked image.
In the first 100000 terms the fixed points are 1,2,15,32,100,115,300,720. It is plausible no more exist although this is unknown.. The sequence is conjectured to be a permutation of the positive integers.
See A354739 for the differences between terms.

			a(9) = 20 as a(8) = 12, and 20 is the smallest unused number that shares a factor with 12 and whose difference from the previous term, 20 - 12 = 8, has not appeared. Note that 10,14,15,16,18 all share a factor with 12 but their differences from 12, namely -2,2,3,4,6, have already appeared as differences between previous pairs of terms.

Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A354739, A354687, A354575, A354679, A354688.

A354575 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

Original entry on oeis.org

1, 2, 5, 3, 7, 4, 9, 8, 15, 11, 6, 17, 10, 19, 13, 21, 23, 12, 25, 16, 31, 14, 33, 20, 37, 18, 41, 26, 47, 22, 49, 27, 43, 29, 35, 53, 24, 55, 28, 57, 34, 59, 38, 71, 30, 67, 32, 73, 36, 79, 39, 61, 45, 77, 46, 81, 91, 40, 87, 44, 83, 50, 99, 52, 97, 42, 95, 51, 65, 89, 63, 101, 48, 103, 54, 113

Offset: 1

Scott R. Shannon, Jun 05 2022

This sequence uses a similar rule to A354688 but here the sign of the difference between a(n-1) and a(n) is considered. This leads to the terms showing much more erratic behavior than A354688; see the linked image.
In the first 200000 terms the fixed points are 1,2,8,35, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
See A354679 for the differences between terms.

			a(9) = 15 as a(8) = 8, and 15 is the smallest unused number that is coprime to 8 and whose difference from the previous term, 15 - 8 = 7, has not appeared. Note that 11 and 13 are coprime to 8 but their differences from 8, namely 3 and 5, have already appeared as differences between previous pairs of terms.
a(15) = 13 as a(14) = 19, and 13 is the smallest unused number that is coprime to 19 and whose difference from the previous term, 13 - 19 = -6, has not appeared. Note that 12 is coprime to 19 and smaller than 13 but its difference from 19, namely -7, has already appeared as a difference between a(13) and a(12).

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.

Cf. A354679, A354688, A354727, A354739, A354687.

A354679 First differences of A354575.

Original entry on oeis.org

1, 3, -2, 4, -3, 5, -1, 7, -4, -5, 11, -7, 9, -6, 8, 2, -11, 13, -9, 15, -17, 19, -13, 17, -19, 23, -15, 21, -25, 27, -22, 16, -14, 6, 18, -29, 31, -27, 29, -23, 25, -21, 33, -41, 37, -35, 41, -37, 43, -40, 22, -16, 32, -31, 35, 10, -51, 47, -43, 39, -33, 49, -47, 45, -55, 53, -44, 14, 24, -26

Offset: 1

Scott R. Shannon, Jun 05 2022

See A354575 for further details.

			a(3) = -2 as A354575(4) - A354575(3) = 3 - 5 = -2.

Scott R. Shannon, Image of the first 200000 terms. The blue line is y = 0.

Cf. A354575, A354688, A354727, A354739, A354687.

cp's OEIS Frontend

A354727 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

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A354575 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

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A354679 First differences of A354575.

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