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.

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, 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, 13, 2, 16, 7, 16, 6, 27, 4, 7, 16, 20, 8, 15, 4, 22

Offset

1,3

Comments

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}.

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.

Links

Table of n, a(n) for n=1..80.

Example

The sequence of antichains begins:
   2: {{1}}
   3: {{1,1}}
   3: {{1},{1}}
   4: {{1,2}}
   5: {{1,1,1}}
   5: {{1},{1},{1}}
   6: {{1,1,2}}
   7: {{1,1,1,1}}
   7: {{1,1},{1,1}}
   7: {{1},{1},{1},{1}}
   8: {{1,2,3}}
   9: {{1,1,2,2}}
  10: {{1,1,1,2}}
  10: {{1,1},{1,2}}
  11: {{1,1,1,1,1}}
  11: {{1},{1},{1},{1},{1}}
  12: {{1,1,2,3}}
  12: {{1,2},{1,3}}
  13: {{1,1,1,1,1,1}}
  13: {{1,1,1},{1,1,1}}
  13: {{1,1},{1,1},{1,1}}
  13: {{1},{1},{1},{1},{1},{1}}
  14: {{1,1,1,1,2}}
  14: {{1,2},{1,1,1}}
  15: {{1,1,1,2,2}}
  15: {{1,1},{1,2,2}}
  16: {{1,2,3,4}}

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[Times@@Prime/@nrmptn[n]], And[zensity[#]==-1, Length[zsm[#]]==1, Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]=={}]&]], {n, 50}]

Crossrefs

Cf. A001055, A007718, A030019, A181821, A293607, A303837, A304382, A305081, A305936, A318284, A321229, A321270, A321271, A321272.

Sequence in context: A239936 A239701 A303755 * A294882 A048220 A182593

Adjacent sequences: A321276 A321277 A321278 * A321280 A321281 A321282

Keyword

nonn

Author

Gus Wiseman, Nov 01 2018

Status

approved