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A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

%I A321698 #20 Dec 28 2018 13:59:26
%S A321698 1,2,3,4,5,7,8,9,11,13,15,16,17,19,23,25,27,29,31,32,33,41,43,47,49,
%T A321698 51,53,55,59,64,67,73,79,81,83,85,93,97,101,103,109,113,121,123,125,
%U A321698 127,128,131,137,139,149,151,155,157,161,163,165,167,169,177,179
%N A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.
%C A321698 A multiset multisystem is a finite multiset of finite multisets. 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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.
%C A321698 A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform and regular, so its MM-number 15463 belongs to the sequence.
%e A321698 The sequence of all uniform regular multiset multisystems, together with their MM-numbers, begins:
%e A321698    1: {}                   33: {{1},{3}}            109: {{10}}
%e A321698    2: {{}}                 41: {{6}}                113: {{1,2,3}}
%e A321698    3: {{1}}                43: {{1,4}}              121: {{3},{3}}
%e A321698    4: {{},{}}              47: {{2,3}}              123: {{1},{6}}
%e A321698    5: {{2}}                49: {{1,1},{1,1}}        125: {{2},{2},{2}}
%e A321698    7: {{1,1}}              51: {{1},{4}}            127: {{11}}
%e A321698    8: {{},{},{}}           53: {{1,1,1,1}}          128: {{},{},{},{},{},{}}
%e A321698    9: {{1},{1}}            55: {{2},{3}}            131: {{1,1,1,1,1}}
%e A321698   11: {{3}}                59: {{7}}                137: {{2,5}}
%e A321698   13: {{1,2}}              64: {{},{},{},{},{},{}}  139: {{1,7}}
%e A321698   15: {{1},{2}}            67: {{8}}                149: {{3,4}}
%e A321698   16: {{},{},{},{}}        73: {{2,4}}              151: {{1,1,2,2}}
%e A321698   17: {{4}}                79: {{1,5}}              155: {{2},{5}}
%e A321698   19: {{1,1,1}}            81: {{1},{1},{1},{1}}    157: {{12}}
%e A321698   23: {{2,2}}              83: {{9}}                161: {{1,1},{2,2}}
%e A321698   25: {{2},{2}}            85: {{2},{4}}            163: {{1,8}}
%e A321698   27: {{1},{1},{1}}        93: {{1},{5}}            165: {{1},{2},{3}}
%e A321698   29: {{1,3}}              97: {{3,3}}              167: {{2,6}}
%e A321698   31: {{5}}               101: {{1,6}}              169: {{1,2},{1,2}}
%e A321698   32: {{},{},{},{},{}}    103: {{2,2,2}}            177: {{1},{7}}
%t A321698 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321698 Select[Range[100],And[SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]
%Y A321698 Cf. A005176, A007016, A112798, A271103, A283877, A299353, A302242, A306017, A319056, A319189, A320324, A321699, A321717, A322554, A322703, A322833.
%K A321698 nonn
%O A321698 1,2
%A A321698 _Gus Wiseman_, Dec 27 2018