A305832 Number of connected components of the n-th FDH set-system.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 1, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1

Offset: 1

Author

Gus Wiseman, Jun 10 2018

Keywords

• nonn

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. The n-th FDH set-system is obtained by repeating this factorization on each index s_i.

Examples

			Let f = A050376. The FD-factorization of 765 is 5*9*17 or f(4)*f(6)*f(10) = f(4)*f(2*3)*f(2*5) with connected components {{{4}},{{2,3},{2,5}}}, so a(765) = 2.
		

Crossrefs

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305831.

Programs

Mathematica

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)]]]]];
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]]]]]]]]];
nn=100;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Length[csm[FDfactor[#]/.FDrules&/@(FDfactor[n]/.FDrules)]],{n,nn}]