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A321188 Number of set systems with no singletons whose multiset union is row n of A305936 (a multiset whose multiplicities are the prime indices of n). 0
1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 11, 0, 0, 0, 4, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,16

COMMENTS

A set system is a finite set of finite nonempty sets.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

LINKS

Table of n, a(n) for n=1..40.

EXAMPLE

The a(36) = 4 set systems with no singletons whose multiset union is {1,1,2,2,3,4}:

  {{1,2},{1,2,3,4}}

  {{1,2,3},{1,2,4}}

  {{1,2},{1,3},{2,4}}

  {{1,2},{1,4},{2,3}}

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

hyp[m_]:=Select[mps[m], And[And@@UnsameQ@@@#, UnsameQ@@#, Min@@Length/@#>1]&];

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

Table[Length[hyp[nrmptn[n]]], {n, 30}]

CROSSREFS

Cf. A000070, A000296, A000569, A050326, A056239, A112798, A283877, A292444, A305936, A306005, A318285, A318361, A320922, A320923, A320924.

Sequence in context: A222399 A222519 A128131 * A322076 A115713 A115633

Adjacent sequences:  A321185 A321186 A321187 * A321189 A321190 A321191

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 29 2018

STATUS

approved

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Last modified October 22 13:15 EDT 2021. Contains 348170 sequences. (Running on oeis4.)