%I #11 Jun 26 2020 11:53:32
%S 1,1,2,3,5,7,11,13,18,21,30,32,43,46,57,64,79,83,103,107,130,141,162,
%T 171,205,214,245,258,297,307,357,373,423,441,493,513,591,607,674,702,
%U 790,817,917,938,1040,1078,1186,1216,1362,1395,1534,1580,1738,1779,1956
%N Number of achiral integer partitions of n.
%C A multiset of multisets is achiral if it is not changed by any permutation of the vertices. An integer partition is achiral if taking the multiset of prime indices of each part gives an achiral multiset of multisets.
%e The a(1) = 1 through a(7) = 13 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (21) (22) (32) (33) (52)
%e (111) (31) (41) (42) (61)
%e (211) (221) (51) (331)
%e (1111) (311) (222) (421)
%e (2111) (321) (511)
%e (11111) (411) (2221)
%e (2211) (3211)
%e (3111) (4111)
%e (21111) (22111)
%e (111111) (31111)
%e (211111)
%e (1111111)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t Table[Length[Select[IntegerPartitions[n],Length[graprms[primeMS/@#]]==1&]],{n,0,30}]
%Y The fully-chiral version is A330228.
%Y The Heinz numbers of these partitions are given by A330232.
%Y Achiral set-systems are counted by A083323.
%Y BII-numbers of achiral set-systems are A330217.
%Y Non-isomorphic achiral multiset partitions are A330223.
%Y Achiral factorizations are A330234.
%Y Cf. A001055, A056239, A112798, A319564, A330098, A330230.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 08 2019
%E More terms from _Jinyuan Wang_, Jun 26 2020