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%I #5 Jun 11 2018 13:13:13
%S 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,3,1,2,1,2,1,3,1,2,2,2,
%T 2,1,1,2,1,3,1,3,1,2,2,2,1,2,1,2,1,2,1,2,2,3,2,2,1,3,1,2,2,2,1,3,1,2,
%U 2,3,1,2,1,2,2,2,2,2,1,2,1,2,1,3,2,2,1
%N Number of connected components of the n-th FDH set-system.
%C Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. The n-th FDH set-system is obtained by repeating this factorization on each index s_i.
%e Let f = A050376. The FD-factorization of 765 is 5*9*17 or f(4)*f(6)*f(10) = f(4)*f(2*3)*f(2*5) with connected components {{{4}},{{2,3},{2,5}}}, so a(765) = 2.
%t FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>1]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t nn=100;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
%t Table[Length[csm[FDfactor[#]/.FDrules&/@(FDfactor[n]/.FDrules)]],{n,nn}]
%Y Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305831.
%K nonn
%O 1,6
%A _Gus Wiseman_, Jun 10 2018
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