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.

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

Author

Scott R. Shannon, Jun 05 2022

Keywords

• nonn
• look

Comments

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.

Examples

			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).
		

Links

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

Crossrefs

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