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A326536 MM-numbers of multiset partitions where every part has the same average.

%I A326536 #6 Jul 13 2019 09:13:10
%S A326536 1,2,3,4,5,7,8,9,11,13,16,17,19,21,23,25,27,29,31,32,37,41,43,47,49,
%T A326536 53,57,59,61,63,64,67,71,73,79,81,83,89,97,101,103,107,109,113,115,
%U A326536 121,125,127,128,131,133,137,139,145,147,149,151,157,159,163,167
%N A326536 MM-numbers of multiset partitions where every part has the same average.
%C A326536 First differs from A322902 in having 145.
%C A326536 These are numbers where each prime index has the same 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 A326536 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 A326536 The sequence of multiset partitions where every part has the same average, preceded by their MM-numbers, begins:
%e A326536    1: {}
%e A326536    2: {{}}
%e A326536    3: {{1}}
%e A326536    4: {{},{}}
%e A326536    5: {{2}}
%e A326536    7: {{1,1}}
%e A326536    8: {{},{},{}}
%e A326536    9: {{1},{1}}
%e A326536   11: {{3}}
%e A326536   13: {{1,2}}
%e A326536   16: {{},{},{},{}}
%e A326536   17: {{4}}
%e A326536   19: {{1,1,1}}
%e A326536   21: {{1},{1,1}}
%e A326536   23: {{2,2}}
%e A326536   25: {{2},{2}}
%e A326536   27: {{1},{1},{1}}
%e A326536   29: {{1,3}}
%e A326536   31: {{5}}
%e A326536   32: {{},{},{},{},{}}
%t A326536 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326536 Select[Range[100],SameQ@@Mean/@primeMS/@primeMS[#]&]
%Y A326536 Cf. A038041, A051293, A112798, A302242, A320324, A326512, A326515, A326520, A326533, A326534, A326535, A326537.
%K A326536 nonn
%O A326536 1,2
%A A326536 _Gus Wiseman_, Jul 12 2019