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 A330060 #5 Dec 05 2019 08:23:11
%S A330060 1,2,3,4,6,7,8,9,12,13,14,15,16,18,19,21,24,26,27,28,30,32,35,36,37,
%T A330060 38,39,42,45,48,49,52,53,54,56,57,60,63,64,70,72,74,76,78,81,84,89,90,
%U A330060 91,95,96,98,104,105,106,108,111,112,113,114,117,120,126,128
%N A330060 MM-numbers of VDD-normalized multisets of multisets.
%C A330060 First differs from A330104 and A330120 in having 35 and lacking 69, with corresponding multisets of multisets 35: {{2},{1,1}} and 69: {{1},{2,2}}.
%C A330060 First differs from A330108 in having 207 and lacking 175, with corresponding multisets of multisets 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
%C A330060 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 is first by length and then lexicographically.
%C A330060 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 A330060 Brute-force: 43287: {{1},{2,3},{2,2,4}}
%C A330060 Lexicographic: 43143: {{1},{2,4},{2,2,3}}
%C A330060 VDD: 15515: {{2},{1,3},{1,1,4}}
%C A330060 MM: 15265: {{2},{1,4},{1,1,3}}
%C A330060 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 A330060 The sequence of all VDD-normalized multisets of multisets together with their MM-numbers begins:
%e A330060 1: 0 21: {1}{11} 49: {11}{11} 84: {}{}{1}{11}
%e A330060 2: {} 24: {}{}{}{1} 52: {}{}{12} 89: {1112}
%e A330060 3: {1} 26: {}{12} 53: {1111} 90: {}{1}{1}{2}
%e A330060 4: {}{} 27: {1}{1}{1} 54: {}{1}{1}{1} 91: {11}{12}
%e A330060 6: {}{1} 28: {}{}{11} 56: {}{}{}{11} 95: {2}{111}
%e A330060 7: {11} 30: {}{1}{2} 57: {1}{111} 96: {}{}{}{}{}{1}
%e A330060 8: {}{}{} 32: {}{}{}{}{} 60: {}{}{1}{2} 98: {}{11}{11}
%e A330060 9: {1}{1} 35: {2}{11} 63: {1}{1}{11} 104: {}{}{}{12}
%e A330060 12: {}{}{1} 36: {}{}{1}{1} 64: {}{}{}{}{}{} 105: {1}{2}{11}
%e A330060 13: {12} 37: {112} 70: {}{2}{11} 106: {}{1111}
%e A330060 14: {}{11} 38: {}{111} 72: {}{}{}{1}{1} 108: {}{}{1}{1}{1}
%e A330060 15: {1}{2} 39: {1}{12} 74: {}{112} 111: {1}{112}
%e A330060 16: {}{}{}{} 42: {}{1}{11} 76: {}{}{111} 112: {}{}{}{}{11}
%e A330060 18: {}{1}{1} 45: {1}{1}{2} 78: {}{1}{12} 113: {123}
%e A330060 19: {111} 48: {}{}{}{}{1} 81: {1}{1}{1}{1} 114: {}{1}{111}
%t A330060 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330060 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 A330060 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 A330060 Select[Range[100],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]
%Y A330060 Equals the image/fixed points of the idempotent sequence A330061.
%Y A330060 A subset of A320456.
%Y A330060 Non-isomorphic multiset partitions are A007716.
%Y A330060 MM-weight is A302242.
%Y A330060 Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A330098, A330102, A330103, A330105.
%Y A330060 Other fixed points:
%Y A330060 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330060 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330060 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330060 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330060 - BII: A330109 (set-systems).
%K A330060 nonn,eigen
%O A330060 1,2
%A A330060 _Gus Wiseman_, Dec 03 2019