cp's OEIS Frontend

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

%I A305831 #6 Jun 11 2018 13:13:06
%S A305831 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 A305831 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 A305831 1,3,1,2,1,2,2,2,2,2,1,2,1,2,1,3,1,2,1
%N A305831 Number of connected components of the strict integer partition with FDH number n.
%C A305831 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 A305831 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 A305831 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 A305831 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 A305831 nn=200;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
%t A305831 Table[Length[zsm[FDfactor[n]/.FDrules]],{n,nn}]
%Y A305831 Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305832.
%K A305831 nonn
%O A305831 1,6
%A A305831 _Gus Wiseman_, Jun 10 2018