This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319564 #14 Nov 09 2018 21:43:57
%S A319564 1,1,2,3,5,7,10,14,21,29,40,53,73,95,128,168,221,282,368,466,599,759,
%T A319564 962,1201,1513,1881,2345,2901,3590,4407,5416,6614,8083,9827,11937,
%U A319564 14442,17458,21021,25299,30347,36363,43438,51843,61705,73384,87054,103149,121949
%N A319564 Number of T_0 integer partitions of n.
%C A319564 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 A319564 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 A319564 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t A319564 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}]
%t A319564 Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&]],{n,20}]
%Y A319564 Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316978, A316979, A316983, A319558, A319616, A319728.
%K A319564 nonn
%O A319564 0,3
%A A319564 _Gus Wiseman_, Sep 23 2018