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A255582
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a(n)=n when n <= 3, otherwise a(n) is the smallest positive number not yet in the sequence such that gcd(a(n), a(n-1)) <= gcd(a(n), a(n-2)) > 1.
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18
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1, 2, 3, 4, 6, 8, 9, 10, 12, 5, 14, 15, 7, 18, 21, 16, 27, 20, 33, 24, 11, 26, 22, 13, 28, 39, 32, 30, 34, 25, 17, 35, 51, 40, 42, 38, 36, 19, 44, 57, 46, 45, 23, 48, 69, 50, 54, 52, 58, 56, 29, 49, 87, 63, 60, 77, 62, 55, 31, 65, 93, 70, 66, 64, 74, 68, 37
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OFFSET
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1,2
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COMMENTS
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This is a permutation of the natural numbers: the proof for A098550 applies with essentially no changes. [Confirmed by N. J. A. Sloane, Feb 27 2015]
For n > 3, all primes first appear in order as composites with one smaller prime (proof similar to that in A098550).
For any given set S of primes, the subsequence consisting of numbers whose prime factors are exactly the primes in S appears in increasing order. For example, if S = {2,3}, 6 appears first, followed by 12, 18, 24, 36, 48, 54, 72, etc.
Appears to be very similar to A064413. Compare the respective inverses A255479 and A064664; see also A255482. Speaking very loosely, the ratio a(n)/n seems to be about 1/2, 1, or 3/2, just as for A064413, although this is a long way from being proved for either sequence. David Applegate points out that this is (presumably) because primes p >= 13 always occur as part of a subsequence 2p X p Y 3p, and subsequences 2p X p Y 5p, 2p X p Y 7p, etc. that produced the extra curves in the graph of A098550 just do not happen. - N. J. A. Sloane, Feb 27 2015, Mar 05 2015.
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LINKS
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David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015.
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MATHEMATICA
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a[n_] := a[n] = If[n<5, n, For[k=5, True, k++, If[FreeQ[Array[a, n-1], k], If[GCD[k, a[n-2]]>1 && GCD[k, a[n-1]] <= GCD[k, a[n-2]], Return[k]]]]];
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PROG
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(Haskell)
import Data.List (delete)
a255582 n = a255582_list !! (n-1)
a255582_list = 1 : 2 : 3 : f 2 3 [4..] where
f u v ws = y : f v y (delete y ws) where
y = head [z | z <- ws, let d = gcd u z, d > 1, gcd v z <= d]
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CROSSREFS
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A255479 is the inverse permutation.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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