A378098 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n)-1).

2, 4, 5, 10, 6, 8, 9, 12, 14, 16, 13, 26, 18, 15, 20, 22, 19, 38, 24, 21, 27, 30, 25, 35, 28, 32, 34, 36, 23, 46, 33, 39, 42, 40, 44, 48, 41, 82, 50, 45, 51, 54, 47, 94, 52, 56, 49, 63, 57, 60, 55, 65, 58, 62, 64, 66, 68, 70, 61, 122, 72, 69, 75, 78, 74, 76, 71, 142, 80, 84, 77, 88, 79, 158, 86, 90, 81, 87, 93, 96, 92, 98, 85, 95, 100, 102, 99, 105, 91, 104

Offset: 1

Author

Scott R. Shannon, Nov 16 2024

Keywords

• nonn

Comments

A self-referencing EKG-type sequence. The terms appear to be concentrated along two lines, a lower straight line where a(n) is slightly above n and an upper straight line where a(n) is approximately 2*n. For larger values of n the later line is composed solely of even semiprimes.

In the first 100000 terms there are no fixed points, and it is likely none exist. In the same range there are only two terms where a(n) is less than n, for a(29) = 23 and a(97) = 89.

The missing numbers are 1, 3, 7, 11, 17, 29, 31, 37, 43, 53, 59, 67, 73, 83, ... .

Examples

			a(2) = 4 as a(1) = 2 and 4 is the smallest unused number that shares a factor with a(2-1) = a(1) = 2.
a(3) = 5 as a(3) is not referenced earlier in the sequence so it is the lowest unused number that does not violate the sharing factor requirement. It cannot be 3 as that would require a(3) = 3 to share a factor with a(3-1) = a(2) = 4, which is does not, and it cannot be 4 as that has already been used.
		

Links

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

Crossrefs

Cf. A064413, A378030, A121053, A027746.