A316268 FDH numbers of connected strict integer partitions.

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, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 92, 97, 101, 103, 107, 108, 109, 111, 113, 115, 117, 119, 121, 124, 127, 129, 131, 135, 137, 139, 144

Offset: 1

Author

Gus Wiseman, Jun 28 2018

Keywords

• nonn

Comments

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).

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.

Examples

			Sequence of connected strict integer partitions begins (1), (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (13), (6,2).
		

Crossrefs

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A299757, A304714, A304716, A305078, A305079, A305829, A305831.

Programs

Mathematica

nn=100;
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)]]]]];
FDrules=MapIndexed[(#1->#2[[1]])&,Array[FDfactor,nn,1,Union]];
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]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[nn],Length[csm[primeMS/@(FDfactor[#]/.FDrules)]]==1&]