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A330194 MM-number of the MM-normalization of the multiset of multisets with MM-number n.

%I A330194 #5 Dec 05 2019 17:42:12
%S A330194 1,2,3,4,3,6,7,8,9,6,3,12,13,14,15,16,3,18,19,12,21,6,7,24,9,26,27,28,
%T A330194 13,30,3,32,15,6,35,36,37,38,39,24,3,42,13,12,45,14,13,48,49,18,15,52,
%U A330194 53,54,15,56,57,26,3,60,37,6,63,64,39,30,3,12,35,70
%N A330194 MM-number of the MM-normalization of the multiset of multisets with MM-number n.
%C A330194 First differs from A330105 at a(35) = 35, A330105(35) = 69.
%C A330194 First differs from A330061 at a(175) = 175, A330061(175) = 207.
%C A330194 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}}.
%C A330194 We define the MM-normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest MM-number.
%C A330194 For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
%C A330194   Brute-force:    43287: {{1},{2,3},{2,2,4}}
%C A330194   Lexicographic:  43143: {{1},{2,4},{2,2,3}}
%C A330194   VDD:            15515: {{2},{1,3},{1,1,4}}
%C A330194   MM:             15265: {{2},{1,4},{1,1,3}}
%H A330194 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent">Idempotence</a>
%F A330194 a(n) <= n.
%t A330194 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330194 mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
%t A330194 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330194 Table[Map[Times@@Prime/@#&,mmnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]
%Y A330194 This sequence is idempotent and its image/fixed points are A330108.
%Y A330194 Non-isomorphic multiset partitions are A007716.
%Y A330194 MM-weight is A302242.
%Y A330194 Cf. A056239, A112798, A317533, A320456, A330061, A330098, A330105.
%Y A330194 Other fixed points:
%Y A330194 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330194 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330194 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330194 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330194 - BII: A330109 (set-systems).
%K A330194 nonn
%O A330194 1,2
%A A330194 _Gus Wiseman_, Dec 05 2019