%I
%S 0,1,1,2,1,12,5,108,145,974,2210,17073,31950,238591
%N Number of prime polyominoes with n cells.
%C We say that a free polyomino is prime if it cannot be tiled by any other free polyomino besides the 1x1 square and itself.
%C The tiling of P must be with a single polyomino, and that single polyomino may not be the unique monomino or P itself. For example, decomposing the Ttetromino into a 3x1 and a 1x1 would use multiple tiles, and this is not permitted.
%C It can be shown that a(n) > 0 for all n >= 4, by considering the polyomino whose cells are at (0,1), (1,1), (0,2), and (x,0) for all x = 0, 1, ..., n4.
%H Cibulis, Liu, and Wainwright, <a href="http://www.paulsalomon.com/uploads/2/8/3/3/28331113/polyomino_number_theory_(i).pdf">Polyomino Number Theory (I)</a>, Crux Mathematicorum, 28(3) (2002), 147150.
%F a(n) = A000105(n) if n is prime.
%e For n = 4, the Ttetromino cannot be decomposed into smaller congruent polyominoes:
%e ++
%e  
%e ++ ++
%e  
%e ++
%e The other four free tetrominoes can, however:
%e ++
%e  
%e   ++
%e    
%e ++   +++ +++
%e         
%e   +++    ++++
%e         
%e ++ ++ +++ +++
%e Thus a(4) = 1.
%Y Cf. A000105, A125759, A213376.
%K nonn,hard,more,nice
%O 0,4
%A _Drake Thomas_, Mar 11 2021
