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 A330097 #6 Dec 05 2019 08:23:29
%S A330097 1,3,7,9,13,15,19,21,27,35,37,39,45,49,53,57,63,81,89,91,95,105,111,
%T A330097 113,117,131,133,135,141,147,151,159,161,165,169,171,183,189,195,207,
%U A330097 223,225,243,245,247,259,265,267,273,281,285,311,315,329,333,339,343
%N A330097 MM-numbers of VDD-normalized multiset partitions.
%C A330097 First differs from A330122 in having 207 and lacking 175, with corresponding multiset partitions 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
%C A330097 A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
%C A330097 We define the VDD (vertex-degrees decreasing) 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, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
%C A330097 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 A330097 Brute-force: 43287: {{1},{2,3},{2,2,4}}
%C A330097 Lexicographic: 43143: {{1},{2,4},{2,2,3}}
%C A330097 VDD: 15515: {{2},{1,3},{1,1,4}}
%C A330097 MM: 15265: {{2},{1,4},{1,1,3}}
%C A330097 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 A330097 The sequence of all VDD-normalized multiset partitions together with their MM-numbers begins:
%e A330097 1: 0 57: {1}{111} 151: {1122}
%e A330097 3: {1} 63: {1}{1}{11} 159: {1}{1111}
%e A330097 7: {11} 81: {1}{1}{1}{1} 161: {11}{22}
%e A330097 9: {1}{1} 89: {1112} 165: {1}{2}{3}
%e A330097 13: {12} 91: {11}{12} 169: {12}{12}
%e A330097 15: {1}{2} 95: {2}{111} 171: {1}{1}{111}
%e A330097 19: {111} 105: {1}{2}{11} 183: {1}{122}
%e A330097 21: {1}{11} 111: {1}{112} 189: {1}{1}{1}{11}
%e A330097 27: {1}{1}{1} 113: {123} 195: {1}{2}{12}
%e A330097 35: {2}{11} 117: {1}{1}{12} 207: {1}{1}{22}
%e A330097 37: {112} 131: {11111} 223: {11112}
%e A330097 39: {1}{12} 133: {11}{111} 225: {1}{1}{2}{2}
%e A330097 45: {1}{1}{2} 135: {1}{1}{1}{2} 243: {1}{1}{1}{1}{1}
%e A330097 49: {11}{11} 141: {1}{23} 245: {2}{11}{11}
%e A330097 53: {1111} 147: {1}{11}{11} 247: {12}{111}
%e A330097 For example, 1155 is the MM-number of {{1},{2},{3},{1,1}}, which is VDD-normalized, so 1155 belongs to the sequence.
%e A330097 On the other hand, 69 is the MM-number of {{1},{2,2}}, but the VDD-normalization is {{2},{1,1}}, so 69 does not belong to the sequence.
%t A330097 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330097 sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
%t A330097 sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
%t A330097 Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]
%Y A330097 Equals the odd terms of A330060.
%Y A330097 A subset of A320634.
%Y A330097 Non-isomorphic multiset partitions are A007716.
%Y A330097 MM-weight is A302242.
%Y A330097 Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330061, A330098, A330102, A330103, A330105.
%Y A330097 Other fixed points:
%Y A330097 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330097 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330097 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330097 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330097 - BII: A330109 (set-systems).
%K A330097 nonn
%O A330097 1,2
%A A330097 _Gus Wiseman_, Dec 04 2019