A387757 a(1) = 1, a(2) = 3; for n > 2, a(n) is the smallest unused positive number that differs from a(n-1) by more than 1 and shares a factor with a(n-1) if a(n-1) < a(n-2), else it is coprime to a(n-1) if a(n-1) > a(n-2).

1, 3, 5, 2, 4, 7, 9, 11, 6, 8, 13, 10, 12, 17, 14, 16, 19, 15, 18, 23, 20, 22, 25, 21, 24, 29, 26, 28, 31, 27, 30, 37, 32, 34, 39, 35, 40, 33, 36, 41, 38, 42, 47, 43, 86, 45, 48, 53, 44, 46, 49, 51, 55, 52, 50, 54, 59, 56, 58, 61, 57, 60, 67, 62, 64

Offset

1,2

Comments

The sequence is composed of two main lines of terms, the upper, sparsely populated line, being even semiprimes, while the lower contains all other terms. The primes do not appear in their natural order, although those out of order are not common.

There are 73 fixed points in the first 500000 terms, these are 1, 16, 22, 28, 34, ..., 1901, 3174, 33376. It is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Example

a(3) = 5 as a(2) = 3 > a(1) = 1, and 5 is the smallest unused number that differs from 3 by more than 1 and is coprime to 3.
a(5) = 4 as a(4) = 2 < a(3) = 5, and 4 is the smallest unused number that differs from 2 by more than 1 and shares a factor with 2.

Crossrefs

Cf. A387231, A384045, A375563, A375564, A373545, A064413.

Sequence in context: A210521 A219249 A203553 * A081964 A082817 A084753

Adjacent sequences: A387754 A387755 A387756 * A387758 A387759 A387760

Keyword

nonn

Author

Scott R. Shannon, Sep 07 2025

Status

approved