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 and regular, so its MM-number 15463 belongs to the sequence.
EXAMPLE
The sequence of all uniform regular multiset multisystems, together with their MM-numbers, begins:
1: {} 33: {{1},{3}} 109: {{10}}
2: {{}} 41: {{6}} 113: {{1,2,3}}
3: {{1}} 43: {{1,4}} 121: {{3},{3}}
4: {{},{}} 47: {{2,3}} 123: {{1},{6}}
5: {{2}} 49: {{1,1},{1,1}} 125: {{2},{2},{2}}
7: {{1,1}} 51: {{1},{4}} 127: {{11}}
8: {{},{},{}} 53: {{1,1,1,1}} 128: {{},{},{},{},{},{}}
9: {{1},{1}} 55: {{2},{3}} 131: {{1,1,1,1,1}}
11: {{3}} 59: {{7}} 137: {{2,5}}
13: {{1,2}} 64: {{},{},{},{},{},{}} 139: {{1,7}}
15: {{1},{2}} 67: {{8}} 149: {{3,4}}
16: {{},{},{},{}} 73: {{2,4}} 151: {{1,1,2,2}}
17: {{4}} 79: {{1,5}} 155: {{2},{5}}
19: {{1,1,1}} 81: {{1},{1},{1},{1}} 157: {{12}}
23: {{2,2}} 83: {{9}} 161: {{1,1},{2,2}}
25: {{2},{2}} 85: {{2},{4}} 163: {{1,8}}
27: {{1},{1},{1}} 93: {{1},{5}} 165: {{1},{2},{3}}
29: {{1,3}} 97: {{3,3}} 167: {{2,6}}
31: {{5}} 101: {{1,6}} 169: {{1,2},{1,2}}
32: {{},{},{},{},{}} 103: {{2,2,2}} 177: {{1},{7}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], And[SameQ@@PrimeOmega/@primeMS[#], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 27 2018
STATUS
approved