%I #14 Nov 09 2018 21:43:57

%S 1,1,2,3,5,7,10,14,21,29,40,53,73,95,128,168,221,282,368,466,599,759,

%T 962,1201,1513,1881,2345,2901,3590,4407,5416,6614,8083,9827,11937,

%U 14442,17458,21021,25299,30347,36363,43438,51843,61705,73384,87054,103149,121949

%N Number of T_0 integer partitions of n.

%C The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).

%C Also the number of integer partitions of n with no equivalent primes. In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent. See A316978 for more examples.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}]

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&]],{n,20}]

%Y Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316978, A316979, A316983, A319558, A319616, A319728.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 23 2018