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, 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.
EXAMPLE
The sequence of all strict uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
1: {}
2: {{}}
3: {{1}}
7: {{1,1}}
13: {{1,2}}
15: {{1},{2}}
19: {{1,1,1}}
53: {{1,1,1,1}}
113: {{1,2,3}}
131: {{1,1,1,1,1}}
151: {{1,1,2,2}}
161: {{1,1},{2,2}}
165: {{1},{2},{3}}
311: {{1,1,1,1,1,1}}
719: {{1,1,1,1,1,1,1}}
1291: {{1,2,3,4}}
1321: {{1,1,1,2,2,2}}
1619: {{1,1,1,1,1,1,1,1}}
1937: {{1,2},{3,4}}
1957: {{1,1,1},{2,2,2}}
2021: {{1,4},{2,3}}
2093: {{1,1},{1,2},{2,2}}
2117: {{1,3},{2,4}}
2257: {{1,1,2},{1,2,2}}
2805: {{1},{2},{3},{4}}
3671: {{1,1,1,1,1,1,1,1,1}}
6997: {{1,1,2,2,3,3}}
8161: {{1,1,1,1,1,1,1,1,1,1}}
10627: {{1,1,1,1,2,2,2,2}}
13969: {{1,2,2},{1,3,3}}
13987: {{1,1,3},{2,2,3}}
14023: {{1,1,2},{2,3,3}}
15617: {{1,1},{2,2},{3,3}}
17719: {{1,2},{1,3},{2,3}}
17863: {{1,1,1,1,1,1,1,1,1,1,1}}
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[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], SameQ@@PrimeOmega/@primeMS[#], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 27 2018
STATUS
approved