OFFSET
1,1
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.
EXAMPLE
Sequence of connected strict integer partitions begins (1), (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (13), (6,2).
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&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 28 2018
STATUS
approved