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A305831 Number of connected components of the strict integer partition with FDH number n. 3


%S 0,1,1,1,1,2,1,2,1,2,1,2,1,2,1,1,1,2,1,2,2,2,1,3,1,2,1,2,1,2,1,2,2,2,

%T 2,1,1,2,1,3,1,3,1,2,1,2,1,2,1,2,1,2,1,2,2,3,2,2,1,2,1,2,2,1,1,3,1,2,

%U 1,3,1,2,1,2,2,2,2,2,1,2,1,2,1,3,1,2,1

%N Number of connected components of the strict integer partition with FDH number n.

%C Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

%e Let f = A050376. The FD-factorization of 1683 is 9*11*17 = f(6)*f(7)*f(10). The connected components of {6,7,10} are {{7},{6,10}}, so a(1683) = 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 zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t nn=200;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];

%t Table[Length[zsm[FDfactor[n]/.FDrules]],{n,nn}]

%Y Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305832.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jun 10 2018

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Last modified November 14 22:45 EST 2019. Contains 329135 sequences. (Running on oeis4.)