Solution below.

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It’s Fibonacci time, again!

I can construct the sets with integers from 1 to 5 by, a) including the sets with integers from 1 to 4 (8 of those), b) taking the sets with integers from 1 to 3 and adding a 5 (5 of those). (Remember that a set including 5 can’t already have 4.) 5+8 = 13. F(6) = 8. So we’re looking for F(16) = 987.