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MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.
3

%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