A378030 -id:A378030 - cp's OEIS frontend

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

Scott R. Shannon, Nov 16 2024

nonn

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, ... .

			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.

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

Cf. A064413, A378030, A121053, A027746.

A378116 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n)-2) while not sharing a factor with a(a(n)-1).

Original entry on oeis.org

3, 4, 9, 8, 7, 6, 35, 12, 25, 11, 15, 22, 14, 33, 16, 21, 18, 49, 20, 63, 26, 27, 19, 24, 95, 28, 45, 32, 31, 30, 217, 34, 77, 36, 55, 38, 39, 40, 51, 44, 42, 121, 46, 99, 50, 57, 43, 48, 215, 52, 75, 56, 54, 91, 58, 65, 62, 85, 60, 119, 64, 105, 68, 69, 70, 61, 71, 122, 213, 74, 81, 73, 78, 365, 76, 115, 82, 125, 83, 80, 249, 86, 87, 88, 93, 92, 111, 94, 84

Offset: 1

Scott R. Shannon, Nov 17 2024

nonn

A self-referencing Yellowstone permutation-like sequence. Care must be taken when finding a(n) to ensure that, if n+1 has previously appeared in the sequence, a(n-1) has at least one prime factor not in a(n), else a(n+1) would not exist.
In the first 100000 terms the fixed points are 6, 24, 30, 48, 80, 240, 629, 2328, 2532, 3960, 17130, 29850, 57480, 57876, 60180. It is likely more exist.
The missing numbers are 1, 2, 5, 10, 13, 17, 23, 29, 37, 41, 47, 53, 59, 66 ... .

			a(1) = 3 as a(3) = 9 and 9 is the smallest unused number that shares a factor with a(3-2) = a(1) = 3 while not sharing a factor with a(3-1) = a(2) = 4.
a(5) = 7 as a(7) = 35 and 35 is the smallest unused number that shares a factor with a(7-2) = a(5) = 7 while not sharing a factor with a(7-1) = a(6) = 6. Note that a(5) cannot be 5 as 5 does not share a factor with a(5-2) = a(3) = 9, nor can it be 6 as that would imply a(6) shares a factor with a(6-2) = a(4) = 8 while not sharing a factor with a(6-1) = a(5) = 6, which is impossible.

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. A098550, A378098, A378030, A064413, A378030, A121053, A027746.

A383069 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n-1)).

Original entry on oeis.org

2, 3, 6, 1, 7, 8, 10, 5, 11, 35, 14, 9, 15, 22, 33, 12, 13, 16, 18, 19, 21, 24, 17, 39, 20, 27, 38, 23, 30, 34, 25, 29, 36, 26, 40, 42, 28, 68, 247, 45, 31, 69, 32, 46, 50, 58, 37, 43, 44, 52, 41, 62, 47, 55, 74, 48, 59, 54, 215, 49, 56, 1147, 51, 65, 82, 53, 70, 92, 145, 94, 57, 73, 118

Offset: 1

Scott R. Shannon, Apr 15 2025

nonn

The majority of the terms are concentrated along a line a(n) ~ 0.91*n, although sparser lines of concentration also exist. Almost all terms are less than 2*n, although there are large outliers that equal the product of two large primes and are much larger than this range.
It is likely 4 is the only positive integer not to appear. Interestingly if we force the sequence to be composed of only terms greater than 1, the only term that differs is a(4), which becomes 4 instead of 1, with all other terms remaining the same.
In the first 10000 terms the only fixed points are 21 and 104. It is unknown if more exist.

			a(1) = 2. The first term cannot be 1 as that would force a(2) to share a factor with a(a(1)) = 1, so the next smallest unused number, 2, is chosen.
a(2) = 3. The second term cannot be 1 as that would force a(a(2)) = a(1) = 1 to share a factor with a(a(1)) = a(1) = 2, so the next smallest unused number, 3, is chosen. This now forces a(a(2)) = a(3) to share a factor with a(a(1)) = a(2) = 3.
a(3) = 6. This is the smallest unused number that is a multiple of 3, which was forced by the previous a(1) = 2 and a(2) = 3.
a(4) = 1. As this term has not been previous referenced, 1 can be chosen. Note this eliminates 4 as a possible term for the remainder of the sequence as a(4) cannot share a factor with any other number. Note that as a(a(4)) = a(1) = 2, a(5) must be an index to an even number.
a(5) = 7. As previously noted, no term can be 4, and a(5) must be an index to an even number, so it cannot be 5 itself. The number 6 has been used, so that leaves 7 as the smallest possible choice. Note this now forces a(7) to share a factor with a(a(4)) = a(1) = 2.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 10000 terms for a(n) <= 20000. The green line is a(n) = n.

Cf. A378098, A064413, A378116, A378030.

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

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A378116 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n)-2) while not sharing a factor with a(a(n)-1).

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A383069 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n-1)).

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