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 A330122 #6 Dec 05 2019 17:41:40
%S A330122 1,3,7,9,13,15,19,21,27,35,37,39,45,49,53,57,63,81,89,91,95,105,111,
%T A330122 113,117,131,133,135,141,147,151,159,161,165,169,171,175,183,189,195,
%U A330122 223,225,243,245,247,259,265,267,273,281,285,311,315,329,333,339,343
%N A330122 MM-numbers of MM-normalized multiset partitions.
%C A330122 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 A330122 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 A330122 Brute-force: 43287: {{1},{2,3},{2,2,4}}
%C A330122 Lexicographic: 43143: {{1},{2,4},{2,2,3}}
%C A330122 VDD: 15515: {{2},{1,3},{1,1,4}}
%C A330122 MM: 15265: {{2},{1,4},{1,1,3}}
%C A330122 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 A330122 The sequence of all MM-normalized multiset partitions together with their MM-numbers begins:
%e A330122 1: 0 57: {1}{111} 151: {1122}
%e A330122 3: {1} 63: {1}{1}{11} 159: {1}{1111}
%e A330122 7: {11} 81: {1}{1}{1}{1} 161: {11}{22}
%e A330122 9: {1}{1} 89: {1112} 165: {1}{2}{3}
%e A330122 13: {12} 91: {11}{12} 169: {12}{12}
%e A330122 15: {1}{2} 95: {2}{111} 171: {1}{1}{111}
%e A330122 19: {111} 105: {1}{2}{11} 175: {2}{2}{11}
%e A330122 21: {1}{11} 111: {1}{112} 183: {1}{122}
%e A330122 27: {1}{1}{1} 113: {123} 189: {1}{1}{1}{11}
%e A330122 35: {2}{11} 117: {1}{1}{12} 195: {1}{2}{12}
%e A330122 37: {112} 131: {11111} 223: {11112}
%e A330122 39: {1}{12} 133: {11}{111} 225: {1}{1}{2}{2}
%e A330122 45: {1}{1}{2} 135: {1}{1}{1}{2} 243: {1}{1}{1}{1}{1}
%e A330122 49: {11}{11} 141: {1}{23} 245: {2}{11}{11}
%e A330122 53: {1111} 147: {1}{11}{11} 247: {12}{111}
%t A330122 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330122 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 A330122 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330122 Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]
%Y A330122 Equals the odd terms of A330108.
%Y A330122 A subset of A320456.
%Y A330122 Non-isomorphic multiset partitions are A007716.
%Y A330122 MM-weight is A302242.
%Y A330122 Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.
%Y A330122 Other fixed points:
%Y A330122 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330122 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330122 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330122 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330122 - BII: A330109 (set-systems).
%K A330122 nonn
%O A330122 1,2
%A A330122 _Gus Wiseman_, Dec 05 2019