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A326537
MM-numbers of multiset partitions where each part has a different average.
11
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110
OFFSET
1,2
COMMENTS
These are numbers where each prime index has a different average of prime indices. 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 obtained by taking the multiset of prime indices of each prime index 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}}.
EXAMPLE
The sequence of multiset partitions where each part has a different average, preceded by their MM-numbers, begins:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
6: {{},{1}}
7: {{1,1}}
10: {{},{2}}
11: {{3}}
13: {{1,2}}
14: {{},{1,1}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
22: {{},{3}}
23: {{2,2}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], UnsameQ@@Mean/@primeMS/@primeMS[#]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2019
STATUS
approved