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A330236 MM-numbers of fully chiral multisets of multisets.

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%I A330236 #7 Dec 10 2019 12:12:15
%S A330236 1,2,3,4,5,6,7,8,9,10,11,12,14,16,17,18,19,20,21,22,23,24,25,27,28,31,
%T A330236 32,34,35,36,37,38,39,40,41,42,44,45,46,48,49,50,53,54,56,57,59,61,62,
%U A330236 63,64,65,67,68,69,70,71,72,74,75,76,77,78,80,81,82,83
%N A330236 MM-numbers of fully chiral multisets of multisets.
%C A330236 A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.
%C A330236 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}}.
%F A330236 Numbers n such that A330098(n) = A303975(n)!.
%e A330236 The sequence of all fully chiral multisets of multisets together with their MM-numbers begins:
%e A330236    1:             18: {}{1}{1}      37: {112}          57: {1}{111}
%e A330236    2: {}          19: {111}         38: {}{111}        59: {7}
%e A330236    3: {1}         20: {}{}{2}       39: {1}{12}        61: {122}
%e A330236    4: {}{}        21: {1}{11}       40: {}{}{}{2}      62: {}{5}
%e A330236    5: {2}         22: {}{3}         41: {6}            63: {1}{1}{11}
%e A330236    6: {}{1}       23: {22}          42: {}{1}{11}      64: {}{}{}{}{}{}
%e A330236    7: {11}        24: {}{}{}{1}     44: {}{}{3}        65: {2}{12}
%e A330236    8: {}{}{}      25: {2}{2}        45: {1}{1}{2}      67: {8}
%e A330236    9: {1}{1}      27: {1}{1}{1}     46: {}{22}         68: {}{}{4}
%e A330236   10: {}{2}       28: {}{}{11}      48: {}{}{}{}{1}    69: {1}{22}
%e A330236   11: {3}         31: {5}           49: {11}{11}       70: {}{2}{11}
%e A330236   12: {}{}{1}     32: {}{}{}{}{}    50: {}{2}{2}       71: {113}
%e A330236   14: {}{11}      34: {}{4}         53: {1111}         72: {}{}{}{1}{1}
%e A330236   16: {}{}{}{}    35: {2}{11}       54: {}{1}{1}{1}    74: {}{112}
%e A330236   17: {4}         36: {}{}{1}{1}    56: {}{}{}{11}     75: {1}{2}{2}
%e A330236 The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.
%t A330236 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330236 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t A330236 Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]
%Y A330236 Costrict (or T_0) factorizations are A316978.
%Y A330236 BII-numbers of fully chiral set-systems are A330226.
%Y A330236 Non-isomorphic fully chiral multiset partitions are A330227.
%Y A330236 Full chiral partitions are A330228.
%Y A330236 Fully chiral covering set-systems are A330229.
%Y A330236 Fully chiral factorizations are A330235.
%Y A330236 Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.
%K A330236 nonn
%O A330236 1,2
%A A330236 _Gus Wiseman_, Dec 10 2019