A093714 -id:A093714 - cp's OEIS frontend

A358277 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous Omega(a(n-1)) terms.

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 10, 13, 12, 17, 14, 15, 19, 16, 23, 18, 25, 29, 20, 21, 31, 22, 27, 35, 26, 33, 37, 24, 41, 28, 43, 30, 47, 32, 53, 34, 39, 49, 38, 45, 59, 36, 61, 40, 67, 42, 71, 44, 65, 51, 46, 55, 57, 52, 73, 48, 79, 50, 77, 69, 58, 83, 54, 85, 89, 56, 97, 60, 101, 62, 63, 95

Offset: 1

Scott R. Shannon, Nov 08 2022

nonn

Unlike A356850 all the terms are concentrated along three straight lines. In the first 100000 terms there are ten fixed points, 1, 2, 3, ..., 27, 57, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(4) = 4 as a(3) = 3 and Omega(3) = A001222(3) = 1, and 4 is coprime to the previous one term, namely a(3) = 3.
a(9) = 11 as a(8) = 8 and Omega(8) = A001222(8) = 3, and 11 is coprime to the previous three terms, namely 8, 7, 6.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^10, with records in red, local minima in blue, highlighting primes in green and other prime powers in gold, showing numbers that are neither squarefree nor prime powers in magenta.
Michael De Vlieger, Scalar scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, composite squarefree numbers in green, products of composite prime powers in magenta, and other numbers in blue.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^10 using a color function showing Omega(a(n-1)).

Cf. A356850, A356903, A001222, A356851, A093714, A336957, A000040.

nn = 90; c[] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 3; j = 2; Do[k = u; m = Times @@ Array[a[n - #] &, PrimeOmega[j]]; While[Nand[! c[k], CoprimeQ[k, m]], k++]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Nov 20 2022 *)

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

A358277 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous Omega(a(n-1)) terms.

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