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%I #18 Dec 28 2018 13:59:39
%S 1,2,3,4,7,8,9,13,15,16,19,27,32,49,53,64,81,113,128,131,151,161,165,
%T 169,225,243,256,311,343,361,512,719,729,1024,1291,1321,1619,1937,
%U 1957,2021,2048,2093,2117,2187,2197,2257,2401,2805,2809,3375,3671,4096,6561
%N MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.
%C 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 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, regular, and spans an initial interval of positive integers, so its MM-number 15463 belongs to the sequence.
%e The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
%e 1: {}
%e 2: {{}}
%e 3: {{1}}
%e 4: {{},{}}
%e 7: {{1,1}}
%e 8: {{},{},{}}
%e 9: {{1},{1}}
%e 13: {{1,2}}
%e 15: {{1},{2}}
%e 16: {{},{},{},{}}
%e 19: {{1,1,1}}
%e 27: {{1},{1},{1}}
%e 32: {{},{},{},{},{}}
%e 49: {{1,1},{1,1}}
%e 53: {{1,1,1,1}}
%e 64: {{},{},{},{},{},{}}
%e 81: {{1},{1},{1},{1}}
%e 113: {{1,2,3}}
%e 128: {{},{},{},{},{},{},{}}
%e 131: {{1,1,1,1,1}}
%e 151: {{1,1,2,2}}
%e 161: {{1,1},{2,2}}
%e 165: {{1},{2},{3}}
%e 169: {{1,2},{1,2}}
%e 225: {{1},{1},{2},{2}}
%e 243: {{1},{1},{1},{1},{1}}
%e 256: {{},{},{},{},{},{},{},{}}
%e 311: {{1,1,1,1,1,1}}
%e 343: {{1,1},{1,1},{1,1}}
%e 361: {{1,1,1},{1,1,1}}
%e 512: {{},{},{},{},{},{},{},{},{}}
%e 719: {{1,1,1,1,1,1,1}}
%e 729: {{1},{1},{1},{1},{1},{1}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t Select[Range[1000],And[normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]
%Y Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 27 2018
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