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}}.
LINKS
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[#]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2019
STATUS
approved