cp's OEIS Frontend

A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

%I A321279 #4 Nov 02 2018 11:23:18
%S A321279 0,1,2,1,2,1,3,1,1,2,2,2,4,2,2,1,2,3,4,4,2,4,3,4,4,3,4,6,4,6,2,1,4,6,
%T A321279 4,9,6,5,3,9,2,8,4,9,8,7,4,8,4,12,6,12,5,16,8,17,5,7,2,19,6,10,10,1,6,
%U A321279 13,2,16,7,16,6,27,4,7,16,20,8,15,4,22
%N A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.
%C A321279 This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%C A321279 The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
%e A321279 The sequence of antichains begins:
%e A321279    2: {{1}}
%e A321279    3: {{1,1}}
%e A321279    3: {{1},{1}}
%e A321279    4: {{1,2}}
%e A321279    5: {{1,1,1}}
%e A321279    5: {{1},{1},{1}}
%e A321279    6: {{1,1,2}}
%e A321279    7: {{1,1,1,1}}
%e A321279    7: {{1,1},{1,1}}
%e A321279    7: {{1},{1},{1},{1}}
%e A321279    8: {{1,2,3}}
%e A321279    9: {{1,1,2,2}}
%e A321279   10: {{1,1,1,2}}
%e A321279   10: {{1,1},{1,2}}
%e A321279   11: {{1,1,1,1,1}}
%e A321279   11: {{1},{1},{1},{1},{1}}
%e A321279   12: {{1,1,2,3}}
%e A321279   12: {{1,2},{1,3}}
%e A321279   13: {{1,1,1,1,1,1}}
%e A321279   13: {{1,1,1},{1,1,1}}
%e A321279   13: {{1,1},{1,1},{1,1}}
%e A321279   13: {{1},{1},{1},{1},{1},{1}}
%e A321279   14: {{1,1,1,1,2}}
%e A321279   14: {{1,2},{1,1,1}}
%e A321279   15: {{1,1,1,2,2}}
%e A321279   15: {{1,1},{1,2,2}}
%e A321279   16: {{1,2,3,4}}
%t A321279 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A321279 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A321279 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 A321279 zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
%t A321279 Table[Length[Select[facs[Times@@Prime/@nrmptn[n]],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]
%Y A321279 Cf. A001055, A007718, A030019, A181821, A293607, A303837, A304382, A305081, A305936, A318284, A321229, A321270, A321271, A321272.
%K A321279 nonn
%O A321279 1,3
%A A321279 _Gus Wiseman_, Nov 01 2018