login
A321698
MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.
4
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 41, 43, 47, 49, 51, 53, 55, 59, 64, 67, 73, 79, 81, 83, 85, 93, 97, 101, 103, 109, 113, 121, 123, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 169, 177, 179
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[#]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 27 2018
STATUS
approved