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A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers. 3

%I

%S 1,2,3,7,13,15,19,53,113,131,151,161,165,311,719,1291,1321,1619,1937,

%T 1957,2021,2093,2117,2257,2805,3671,6997,8161,10627,13969,13987,14023,

%U 15617,17719,17863,20443,22207,22339,38873,79349,84017,86955,180503,202133

%N Squarefree MM-numbers of strict uniform regular multiset systems 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, 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 The sequence of all strict 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 7: {{1,1}}

%e 13: {{1,2}}

%e 15: {{1},{2}}

%e 19: {{1,1,1}}

%e 53: {{1,1,1,1}}

%e 113: {{1,2,3}}

%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 311: {{1,1,1,1,1,1}}

%e 719: {{1,1,1,1,1,1,1}}

%e 1291: {{1,2,3,4}}

%e 1321: {{1,1,1,2,2,2}}

%e 1619: {{1,1,1,1,1,1,1,1}}

%e 1937: {{1,2},{3,4}}

%e 1957: {{1,1,1},{2,2,2}}

%e 2021: {{1,4},{2,3}}

%e 2093: {{1,1},{1,2},{2,2}}

%e 2117: {{1,3},{2,4}}

%e 2257: {{1,1,2},{1,2,2}}

%e 2805: {{1},{2},{3},{4}}

%e 3671: {{1,1,1,1,1,1,1,1,1}}

%e 6997: {{1,1,2,2,3,3}}

%e 8161: {{1,1,1,1,1,1,1,1,1,1}}

%e 10627: {{1,1,1,1,2,2,2,2}}

%e 13969: {{1,2,2},{1,3,3}}

%e 13987: {{1,1,3},{2,2,3}}

%e 14023: {{1,1,2},{2,3,3}}

%e 15617: {{1,1},{2,2},{3,3}}

%e 17719: {{1,2},{1,3},{2,3}}

%e 17863: {{1,1,1,1,1,1,1,1,1,1,1}}

%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[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

%Y Cf. A005117, A007016, A112798, A302242, A306017, A306021, A319056, A319189, A319190, A320324, A321698, A321699, A322554, A322833.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 27 2018

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Last modified March 29 18:31 EDT 2020. Contains 333117 sequences. (Running on oeis4.)