A249167 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).

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, 36, 30, 44, 32, 52, 16, 40, 48, 34, 57, 17, 19, 51, 38, 60, 46, 35, 23, 42, 92, 50, 64, 25, 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

Offset: 1

Author

Vladimir Shevelev, Dec 15 2014

Keywords

• nonn

Comments

Fermi-Dirac analog of A098550. Recall that every positive digit has a unique Fermi-Dirac representation as a product of distinct terms of A050376.

Conjecture: the sequence is a permutation of the positive integers.

Conjecture is true. The proof is similar to that for A098550 with minor changes. - Vladimir Shevelev, Jan 26 2015

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

Examples

			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.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000
• 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 and J. Int. Seq. 18 (2015) 15.6.7..
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A098550, A050376.

Cf. A213925, A255940 (inverse).

Programs

Haskell

import Data.List (delete, intersect)
a249167 n = a249167_list !! (n-1)
a249167_list = 1 : 2 : 3 : f 2 3 [4..] where
   f u v ws = g ws where
     g (x:xs) | null (intersect fdx $ a213925_row u) ||
                not (null $ intersect fdx $ a213925_row v) = g xs
              | otherwise =  x : f v x (delete x ws)
              where fdx = a213925_row x
-- Reinhard Zumkeller, Mar 11 2015

Extensions

More terms from Peter J. C. Moses, Dec 15 2014