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A351625
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a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with the largest previous term.
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5
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1, 2, 4, 6, 3, 8, 10, 5, 12, 9, 14, 7, 16, 18, 15, 20, 22, 11, 24, 21, 26, 13, 28, 30, 25, 27, 32, 34, 17, 36, 33, 38, 19, 40, 35, 42, 39, 44, 46, 23, 48, 45, 50, 52, 54, 51, 56, 49, 58, 29, 60, 55, 57, 62, 31, 64, 66, 63, 68, 70, 65, 72, 69, 74, 37, 76, 78, 75, 80, 82, 41, 84, 77, 81, 86, 43, 88
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OFFSET
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1,2
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COMMENTS
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The sequence differs from A352976 after twenty-six terms. See the examples below.
In the first 100000 terms the maximum run of even and odd terms is three and five respectively; it is unknown if these runs have a maximum number of terms or are unbounded. The fixed points beyond 2 are 15, 25, 35, and it is likely no more exist. The primes appear in their natural order, and it is conjectured that this is a permutation of the positive integers.
Let u be the smallest missing number in a(1..n-1) and let record r = max(a(1..n-1)).
Theorem: For odd prime q, r = mq -> q. Proof: generally, q may either divide or be coprime to r, but since equality and coprimality are prohibited by definition, primes may only divide records. This implies q serves as local minima, hence u > 2 is always an odd prime and r > 1 is always even.
Since r is always even, r is nondecreasing and even numbers appear in natural order. Also, r = 2q -> q, similar to the Lagarias-Rains-Sloane chain in the EKG sequence but without the successor 3q.
Theorem: a(n) = k such that k is odd iff least prime factor q | k and q | r. Proof: sequence definition demands smallest missing number such that (r,k) > 1. Otherwise, a(n) = a(n-1) + 2, since (a(n-1), a(n-1) + 2) > 1, and all smaller even numbers have already appeared.
All even numbers and primes appear, and every odd composite k has a least prime factor that admits k into the sequence on the occasion of (r,k) > 1.
There are 3 trajectories in scatterplot for a(n) > 1. The trajectory with lowest apparent slope is that of the primes, i.e., local minima. The trajectory with highest slope is that of records, i.e., even numbers, and the remaining numbers are odd composites. (End)
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LINKS
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EXAMPLE
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a(6) = 8 as the largest value of all previous terms is a(4) = 6, and 8 is the smallest unused number that shares a factor with 6.
a(27) = 32 as the largest value of all previous terms is a(24) = 30, and 32 is the smallest unused number that shares a factor with 30. Note that A352976(27) = 33 as it must share a factor with max(25,27) = 27.
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MATHEMATICA
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nn = 120; c[_] = 0; Array[Set[{a[#], c[#]}, {2^(# - 1), #}] &, 3]; u = 3; Do[k = u; While[Nand[c[k] == 0, ! CoprimeQ[k, r]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k > r, r = k]; If[k == u, While[c[u] > 0, u++]], {i, L + 1, nn}]; Array[a, nn] (* Michael De Vlieger, May 08 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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