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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
Sequence in context: A032843 A023774 A319837 * A349759 A050271 A322742
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 27 2018
STATUS
approved

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Last modified July 12 19:58 EDT 2024. Contains 374252 sequences. (Running on oeis4.)