A089088 -id:A089088 - cp's OEIS frontend

A374612 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1) but has a common factor with any other previous term that is also coprime to a(n-1).

1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 7, 10, 21, 16, 27, 20, 33, 26, 11, 12, 35, 13, 18, 49, 22, 39, 28, 45, 32, 51, 38, 17, 19, 24, 55, 34, 57, 40, 63, 44, 65, 36, 77, 30, 91, 46, 75, 23, 42, 85, 48, 95, 52, 69, 50, 81, 56, 87, 62, 29, 31, 54, 115, 58, 93, 64, 99, 68, 105, 74, 117, 37, 60, 119, 66

Offset: 1

Scott R. Shannon, Jul 14 2024

nonn

The terms appear to follow a pattern similar to the A064413 and A373390, i.e., the terms are concentrated along just three lines of different gradient, and the lower line consists only of primes. In the first 10000 terms the primes appear in their natural order, and the fixed points are 1, 2, 3, 4, 28, 98; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(11) = 25 as 25 is coprime to a(10) = 6 while sharing a factor with a(9) = 5, which is itself coprime to a(10) = 6. Note that all other previous terms, other than a(1) = 1, do share a factor with a(10) = 6. This is the first term to differ from A373390.

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

Cf. A373390, A089088, A373998, A098550, A247942.

A353905 a(1) = 1; for n > 1, a(n) = smallest positive number that has not appeared that has a common factor with a(n-1) + the smallest unseen positive number.

Original entry on oeis.org

1, 3, 5, 7, 6, 2, 4, 8, 17, 10, 19, 12, 9, 14, 15, 13, 16, 18, 29, 20, 31, 21, 22, 11, 24, 47, 25, 26, 28, 27, 30, 53, 32, 33, 34, 36, 59, 38, 61, 35, 40, 39, 42, 45, 44, 67, 46, 23, 48, 50, 51, 52, 89, 49, 43, 54, 56, 57, 58, 55, 60, 97, 62, 63, 64, 101, 66, 103, 65, 68, 69, 70, 107, 72, 109

Offset: 1

Scott R. Shannon, May 10 2022

nonn

The sequence is conjectured to be a permutation of the positive integers. In the first 100000 terms the number 18869 holds the record for the greatest number of terms for which it is the lowest unseen number, 4769 terms in all. In the same range there are fifteen fixed points, the last being a(1204), and it is likely no more exist.

			a(2) = 3 as a(1) = 1, the smallest unseen positive number is 2, and 1 + 2 = 3, and 3 is the smallest number that has not yet appeared that shares a factor with 3.
a(5) = 6 as a(4) = 7, the smallest unseen positive number is 2, and 7 + 2 = 9, and 6 is the smallest number that has not yet appeared that shares a factor with 9.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue. Primes and fixed points are highlighted in green and gold, respectively.
Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A337136, A089088, A064413, A085229, A093714.

nn = 120; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, ! CoprimeQ[#, k]], k++] &[a[i - 1] + u]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 15 2022 *)

A374645 a(1) = 1, a(2) = 6, a(3) = 2; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1) but has a common factor with any other previous term that has a common factor with a(n-1).

Original entry on oeis.org

1, 6, 2, 3, 4, 9, 8, 15, 14, 27, 5, 12, 7, 10, 21, 16, 25, 18, 35, 22, 39, 20, 11, 24, 13, 30, 49, 26, 33, 28, 45, 32, 51, 38, 55, 34, 19, 36, 17, 40, 57, 44, 63, 46, 65, 42, 23, 48, 77, 50, 69, 52, 75, 56, 81, 58, 85, 54, 29, 60, 91, 62, 87, 64, 31, 66, 95, 68, 93, 70, 99, 74, 105, 76, 37

Offset: 1

Scott R. Shannon, Jul 15 2024

nonn

The terms appear to follow a pattern similar to the A064413 and A373390, i.e., the terms are concentrated along just three lines of different gradient, and the lower line consists only of primes. In the first 10000 terms the primes appear in their natural order except for a(37) = 19 and a(39) = 17 which are reversed. Many fixed points exist, 2228 in the first 10000 terms, these beginning 1, 12, 16, 18, 24, 32, 40, 48, ... . The sequence is conjectured to be a permutation of the positive integers.

			a(11) = 5 as 5 is coprime to a(10) = 27 while sharing a factor with a(8) = 15, which itself shares a factor with a(10) = 27. This is also the first term that uses a term other than a(2) = 6 as the previous term with which is shares a factor with.

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

Cf. A374612, A373390, A089088, A373998, A098550, A247942.

cp's OEIS Frontend

A374612 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1) but has a common factor with any other previous term that is also coprime to a(n-1).

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A353905 a(1) = 1; for n > 1, a(n) = smallest positive number that has not appeared that has a common factor with a(n-1) + the smallest unseen positive number.

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A374645 a(1) = 1, a(2) = 6, a(3) = 2; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1) but has a common factor with any other previous term that has a common factor with a(n-1).

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