|
%I #5 Jun 29 2018 11:35:40
%S 2,3,4,5,7,9,11,13,15,16,17,19,23,25,27,29,31,36,37,39,41,43,45,47,49,
%T 51,53,59,61,64,65,67,69,71,73,79,81,83,85,87,89,92,97,101,103,107,
%U 108,109,111,113,115,117,119,121,124,127,129,131,135,137,139,144
%N FDH numbers of connected strict integer partitions.
%C Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).
%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 greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set or strict partition S is said to be connected if G(S) is a connected graph.
%e Sequence of connected strict integer partitions begins (1), (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (13), (6,2).
%t nn=100;
%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 FDrules=MapIndexed[(#1->#2[[1]])&,Array[FDfactor,nn,1,Union]];
%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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[nn],Length[csm[primeMS/@(FDfactor[#]/.FDrules)]]==1&]
%Y Cf. A048143, A050376, A064547, A213925, A299755, A299756, A299757, A304714, A304716, A305078, A305079, A305829, A305831.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 28 2018
|
