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A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

%I A322703 #15 Dec 28 2018 13:48:53
%S A322703 1,2,3,7,13,15,19,53,113,131,151,161,165,311,719,1291,1321,1619,1937,
%T A322703 1957,2021,2093,2117,2257,2805,3671,6997,8161,10627,13969,13987,14023,
%U A322703 15617,17719,17863,20443,22207,22339,38873,79349,84017,86955,180503,202133
%N A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.
%C A322703 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 A322703 A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,1},{2,3},{2,3}} is uniform and regular but not strict, so its MM-number 15463 does not belong to the sequence. Note that the parts of parts such as {1,1} do not have to be distinct, only the multiset of parts.
%e A322703 The sequence of all strict uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
%e A322703       1: {}
%e A322703       2: {{}}
%e A322703       3: {{1}}
%e A322703       7: {{1,1}}
%e A322703      13: {{1,2}}
%e A322703      15: {{1},{2}}
%e A322703      19: {{1,1,1}}
%e A322703      53: {{1,1,1,1}}
%e A322703     113: {{1,2,3}}
%e A322703     131: {{1,1,1,1,1}}
%e A322703     151: {{1,1,2,2}}
%e A322703     161: {{1,1},{2,2}}
%e A322703     165: {{1},{2},{3}}
%e A322703     311: {{1,1,1,1,1,1}}
%e A322703     719: {{1,1,1,1,1,1,1}}
%e A322703    1291: {{1,2,3,4}}
%e A322703    1321: {{1,1,1,2,2,2}}
%e A322703    1619: {{1,1,1,1,1,1,1,1}}
%e A322703    1937: {{1,2},{3,4}}
%e A322703    1957: {{1,1,1},{2,2,2}}
%e A322703    2021: {{1,4},{2,3}}
%e A322703    2093: {{1,1},{1,2},{2,2}}
%e A322703    2117: {{1,3},{2,4}}
%e A322703    2257: {{1,1,2},{1,2,2}}
%e A322703    2805: {{1},{2},{3},{4}}
%e A322703    3671: {{1,1,1,1,1,1,1,1,1}}
%e A322703    6997: {{1,1,2,2,3,3}}
%e A322703    8161: {{1,1,1,1,1,1,1,1,1,1}}
%e A322703   10627: {{1,1,1,1,2,2,2,2}}
%e A322703   13969: {{1,2,2},{1,3,3}}
%e A322703   13987: {{1,1,3},{2,2,3}}
%e A322703   14023: {{1,1,2},{2,3,3}}
%e A322703   15617: {{1,1},{2,2},{3,3}}
%e A322703   17719: {{1,2},{1,3},{2,3}}
%e A322703   17863: {{1,1,1,1,1,1,1,1,1,1,1}}
%t A322703 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322703 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A322703 Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]
%Y A322703 Cf. A005117, A007016, A112798, A302242, A306017, A306021, A319056, A319189, A319190, A320324, A321698, A321699, A322554, A322833.
%K A322703 nonn
%O A322703 1,2
%A A322703 _Gus Wiseman_, Dec 27 2018