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%I A000341
%S A000341 1,2,3,6,26,96,210,1106,3759,12577,74072,423884,2333828,16736611
%N A000341 Ways to pair up {1..2n} so sum of each pair is prime.
%H A000341 L. E. Greenfield and S. J. Greenfield, <a 
href="http://www.math.uwaterloo.ca/JIS/index.html#P98.1.2">Some Pr\
  oblems of Combinatorial Number Theory Related to Bertrand's Postulate</a>, J. 
Integer Sequences, 1998, #98.1.2.
%K A000341 nonn,hard,nice
%O A000341 1,2
%A A000341 greenfie at math.rutgers.edu (S. J. Greenfield)
%E A000341 More terms from David W. Wilson (davidwwilson at attbi.com).

Is it true that for k>=2 there always are at least two disjoint decompositions of 
{1,2,...2k} of that kind? Is there a simple proof of this fact?

Example: for k=2 {{1,2},{3,4}} and {{1,4},{2,3}} are the two only possible 
decompositions and they are disjoint. for k=3 {{1,4},{2,3},{5,6}} and 
{{1,6},{2,5},{3,4}} are disjoint while {{1,2},{3,4},{5,6}} intersects each of them.

Someone calculated the greatest cardinality of a set of disjoint decompositions of 
{1,2,...,2k} of that kind?


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