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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

Table of n, a(n) for n=1..53.

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.)