

A255582


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(n1)) <= gcd(a(n), a(n2)) > 1.


18



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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.


LINKS

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.


MATHEMATICA

a[n_] := a[n] = If[n<5, n, For[k=5, True, k++, If[FreeQ[Array[a, n1], k], If[GCD[k, a[n2]]>1 && GCD[k, a[n1]] <= GCD[k, a[n2]], Return[k]]]]];


PROG

(Haskell)
import Data.List (delete)
a255582 n = a255582_list !! (n1)
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]


CROSSREFS

A255479 is the inverse permutation.


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



