A366908 -id:A366908 - cp's OEIS frontend

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

Scott R. Shannon, Oct 29 2023

nonn

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.

			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.

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.

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

A366909 Lexicographically earliest infinite sequence of distinct positive integers such that, for n > 2, a(n) shares a factor with a(n-1) but not with n.

Original entry on oeis.org

1, 5, 10, 15, 21, 35, 20, 25, 55, 33, 30, 65, 40, 85, 17, 51, 39, 13, 26, 91, 52, 117, 42, 7, 14, 49, 70, 45, 57, 19, 38, 95, 50, 75, 66, 11, 22, 77, 28, 63, 60, 115, 23, 69, 161, 105, 56, 119, 34, 187, 44, 99, 78, 143, 104, 169, 130, 125, 110, 121, 88, 165, 80, 135, 84, 133, 76, 171, 152, 209

Offset: 1

Scott R. Shannon, Oct 27 2023

nonn

To ensure the sequence is infinite a(n) must be chosen so that it has at least one distinct prime factor that is not a factor of n+1. The first time this rule is required is when determining a(5); see the examples below. It also does not allow a(2) to equal 3 as that would then share its only prime factor with n = 3. As 2 and 4 share a factor with n = 2, this leaves a(2) = 5 as the first valid value.
One can easily show that no 3-smooth number, see A003586, can be a term; these are all blocked by the requirement that a(n) shares no factor with n, else are blocked as such a choice would violate this condition when choosing a(n+1).
For the terms studied beyond the prime a(855) = 277 all subsequent primes appear in their natural order. The earlier primes 7, 11, 13, 17, 19, 197, 199, 211, 223, 277, 281 are either out of order or reversed. The behavior of prime ordering for larger values of n is unknown.

			a(4) = 15 as 15 does not share a factor with 4 while sharing the factor 5 with a(3) = 10.
a(5) = 21 as 21 does not share a factor with 5 while sharing the factor 3 with a(4) = 15. Note that 3 is unused and satisfies these requirements but as 5 + 1 = 6 = 2*3 contains 3 as a prime factor, a(5) cannot contain 3 as its only distinct prime factor else a(6) would not exist. Likewise a(5) cannot equal 6, 9, 12 or 18.

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

Cf. A366952, A366908, A003586, A159193, A093714, A085229, A119018, A000027.

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs