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 A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers. 3
 1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 49, 53, 64, 81, 113, 128, 131, 151, 161, 165, 169, 225, 243, 256, 311, 343, 361, 512, 719, 729, 1024, 1291, 1321, 1619, 1937, 1957, 2021, 2048, 2093, 2117, 2187, 2197, 2257, 2401, 2805, 2809, 3375, 3671, 4096, 6561 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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}}. 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. LINKS EXAMPLE The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:     1: {}     2: {{}}     3: {{1}}     4: {{},{}}     7: {{1,1}}     8: {{},{},{}}     9: {{1},{1}}    13: {{1,2}}    15: {{1},{2}}    16: {{},{},{},{}}    19: {{1,1,1}}    27: {{1},{1},{1}}    32: {{},{},{},{},{}}    49: {{1,1},{1,1}}    53: {{1,1,1,1}}    64: {{},{},{},{},{},{}}    81: {{1},{1},{1},{1}}   113: {{1,2,3}}   128: {{},{},{},{},{},{},{}}   131: {{1,1,1,1,1}}   151: {{1,1,2,2}}   161: {{1,1},{2,2}}   165: {{1},{2},{3}}   169: {{1,2},{1,2}}   225: {{1},{1},{2},{2}}   243: {{1},{1},{1},{1},{1}}   256: {{},{},{},{},{},{},{},{}}   311: {{1,1,1,1,1,1}}   343: {{1,1},{1,1},{1,1}}   361: {{1,1,1},{1,1,1}}   512: {{},{},{},{},{},{},{},{},{}}   719: {{1,1,1,1,1,1,1}}   729: {{1},{1},{1},{1},{1},{1}} MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]]; Select[Range[1000], And[normQ[primeMS/@primeMS[#]], SameQ@@PrimeOmega/@primeMS[#], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&] CROSSREFS Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833. Sequence in context: A032843 A023774 A319837 * A050271 A322742 A307561 Adjacent sequences:  A321696 A321697 A321698 * A321700 A321701 A321702 KEYWORD nonn AUTHOR Gus Wiseman, Dec 27 2018 STATUS approved

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Last modified April 3 04:21 EDT 2020. Contains 333195 sequences. (Running on oeis4.)