A366952 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with n but does not equal n, and shares a factor with a(n-1).

1, 4, 6, 2, 10, 8, 14, 12, 3, 15, 33, 9, 39, 18, 20, 22, 34, 16, 38, 24, 27, 30, 46, 26, 40, 28, 21, 7, 203, 35, 155, 50, 36, 32, 42, 44, 74, 48, 45, 5, 205, 60, 86, 52, 54, 56, 94, 58, 70, 25, 75, 65, 265, 80, 66, 62, 72, 64, 118, 68, 122, 76, 57, 78, 13, 104, 134, 82, 84, 49, 497, 63, 219

Offset: 1

Author

Scott R. Shannon, Oct 29 2023

Keywords

• nonn

Comments

The sequence is conjectured to be a permutation of the positive integers, although the primes typically take many terms to appear, e.g., a(95890) = 223. When a prime does appear it is often followed by a term that is significantly larger than the average-sized term. See the examples below. The primes do not occur in their natural order.

Examples

			a(3) = 6 as 6 does not equal 3, shares the factor 3 with 3 while sharing the factor 2 with a(2) = 4.
a(29) = 203 as 203 does not equal 29, shares the factor 29 with 29 while sharing the factor 7 with a(28) = 7. This is an example of both n and a(n-1) being primes which forces a(n) to be significantly larger than the average-sized term.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 100000 terms with a(n) < 300000. The green line is a(n) = n.

Crossrefs

Cf. A366908, A159193, A093714, A085229, A119018, A000027.