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%I #8 Mar 26 2020 20:41:50
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T 27,28,29,30,31,32,33,34,36,38,40,41,42,43,44,46,47,48,49,50,51,52,53,
%U 54,55,56,57,58,59,60,62,63,64,66,67,68,72,73,76,79,80
%N MM-numbers of achiral multisets of multisets.
%C First differs from A322554 in lacking 141.
%C A multiset of multisets is achiral if it is not changed by any permutation of the vertices.
%C 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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%e The sequence of non-achiral multisets of multisets (the complement of this sequence) together with their MM-numbers begins:
%e 35: {{2},{1,1}}
%e 37: {{1,1,2}}
%e 39: {{1},{1,2}}
%e 45: {{1},{1},{2}}
%e 61: {{1,2,2}}
%e 65: {{2},{1,2}}
%e 69: {{1},{2,2}}
%e 70: {{},{2},{1,1}}
%e 71: {{1,1,3}}
%e 74: {{},{1,1,2}}
%e 75: {{1},{2},{2}}
%e 77: {{1,1},{3}}
%e 78: {{},{1},{1,2}}
%e 87: {{1},{1,3}}
%e 89: {{1,1,1,2}}
%e 90: {{},{1},{1},{2}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]]
%t Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==1&]
%Y The fully-chiral version is A330236.
%Y Achiral set-systems are counted by A083323.
%Y MG-numbers of planted achiral trees are A214577.
%Y MM-weight is A302242.
%Y MM-numbers of costrict (or T_0) multisets of multisets are A322847.
%Y BII-numbers of achiral set-systems are A330217.
%Y Non-isomorphic achiral multiset partitions are A330223.
%Y Achiral integer partitions are counted by A330224.
%Y Achiral factorizations are A330234.
%Y Cf. A001055, A003238, A007716, A056239, A112798, A303975, A330098, A330230, A330233.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 08 2019
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