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%I #40 May 15 2018 16:39:42
%S 1,2,3,8,15,4,5,12,10,21,18,7,6,28,22,20,11,24,55,14,33,26,27,13,9,39,
%T 36,30,44,32,52,16,40,48,34,57,17,19,51,38,60,46,35,23,42,92,50,64,25,
%U 56,75,58,69,29,54,116,45,68,63,76,70,100,62,84,31,66,124,74,93,37,78,148,65,72,80
%N a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common Fermi-Dirac factor with a(n-2), but none with a(n-1).
%C Fermi-Dirac analog of A098550. Recall that every positive digit has a unique Fermi-Dirac representation as a product of distinct terms of A050376.
%C Conjecture: the sequence is a permutation of the positive integers.
%C Conjecture is true. The proof is similar to that for A098550 with minor changes. - _Vladimir Shevelev_, Jan 26 2015
%C It is interesting that while the first 10000 points (n, A098550(n)) lie on about 8 roughly straight lines, the first 10000 points (n,a(n)) here lie on only about 6 lines (cf. scatterplots of these sequences). - _Vladimir Shevelev_, Jan 26 2015
%H Peter J. C. Moses, <a href="/A249167/b249167.txt">Table of n, a(n) for n = 1..10000</a>
%H David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, <a href="http://arxiv.org/abs/1501.01669">The Yellowstone Permutation</a>, arXiv preprint arXiv:1501.01669, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Sloane/sloane9.html">J. Int. Seq. 18 (2015) 15.6.7</a>..
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e a(4) is not 4, since 2 and 4 have no common Fermi-Dirac divisor; it is not 6, since a(3)=3 and 6 have the common divisor 3. So, a(4)=8, having the Fermi-Dirac representation 8=2*4.
%o (Haskell)
%o import Data.List (delete, intersect)
%o a249167 n = a249167_list !! (n-1)
%o a249167_list = 1 : 2 : 3 : f 2 3 [4..] where
%o f u v ws = g ws where
%o g (x:xs) | null (intersect fdx $ a213925_row u) ||
%o not (null $ intersect fdx $ a213925_row v) = g xs
%o | otherwise = x : f v x (delete x ws)
%o where fdx = a213925_row x
%o -- _Reinhard Zumkeller_, Mar 11 2015
%Y Cf. A098550, A050376.
%Y Cf. A213925, A255940 (inverse).
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Dec 15 2014
%E More terms from _Peter J. C. Moses_, Dec 15 2014
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