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A326537 MM-numbers of multiset partitions where each part has a different average.

%I A326537 #8 Jul 13 2019 14:39:25
%S A326537 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,
%T A326537 41,43,46,47,51,53,55,58,59,61,62,65,66,67,69,70,71,73,74,77,78,79,82,
%U A326537 83,85,86,87,89,91,93,94,95,97,101,102,103,106,107,109,110
%N A326537 MM-numbers of multiset partitions where each part has a different average.
%C A326537 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}}.
%H A326537 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e A326537 The sequence of multiset partitions where each part has a different average, preceded by their MM-numbers, begins:
%e A326537    1: {}
%e A326537    2: {{}}
%e A326537    3: {{1}}
%e A326537    5: {{2}}
%e A326537    6: {{},{1}}
%e A326537    7: {{1,1}}
%e A326537   10: {{},{2}}
%e A326537   11: {{3}}
%e A326537   13: {{1,2}}
%e A326537   14: {{},{1,1}}
%e A326537   15: {{1},{2}}
%e A326537   17: {{4}}
%e A326537   19: {{1,1,1}}
%e A326537   22: {{},{3}}
%e A326537   23: {{2,2}}
%e A326537   26: {{},{1,2}}
%e A326537   29: {{1,3}}
%e A326537   30: {{},{1},{2}}
%e A326537   31: {{5}}
%e A326537   33: {{1},{3}}
%t A326537 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326537 Select[Range[100],UnsameQ@@Mean/@primeMS/@primeMS[#]&]
%Y A326537 Cf. A038041, A051293, A112798, A302242, A320324, A326513, A326516, A326521, A326533, A326534, A326535, A326536.
%K A326537 nonn
%O A326537 1,2
%A A326537 _Gus Wiseman_, Jul 12 2019