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A326533 MM-numbers of multiset partitions where each part has a different length.

%I A326533 #8 Jul 13 2019 09:12:46
%S A326533 1,2,3,5,6,7,10,11,13,14,17,19,21,22,23,26,29,31,34,35,37,38,39,41,42,
%T A326533 43,46,47,53,57,58,59,61,62,65,67,69,70,71,73,74,77,78,79,82,83,86,87,
%U A326533 89,94,95,97,101,103,106,107,109,111,113,114,115,118,119,122
%N A326533 MM-numbers of multiset partitions where each part has a different length.
%C A326533 These are numbers where each prime index has a different Omega (number of prime factors counted with multiplicity). 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 A326533 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 A326533 The sequence of multiset partitions where each part has a different average preceded by their MM-numbers begins:
%e A326533    1: {}
%e A326533    2: {{}}
%e A326533    3: {{1}}
%e A326533    5: {{2}}
%e A326533    6: {{},{1}}
%e A326533    7: {{1,1}}
%e A326533   10: {{},{2}}
%e A326533   11: {{3}}
%e A326533   13: {{1,2}}
%e A326533   14: {{},{1,1}}
%e A326533   17: {{4}}
%e A326533   19: {{1,1,1}}
%e A326533   21: {{1},{1,1}}
%e A326533   22: {{},{3}}
%e A326533   23: {{2,2}}
%e A326533   26: {{},{1,2}}
%e A326533   29: {{1,3}}
%e A326533   31: {{5}}
%e A326533   34: {{},{4}}
%e A326533   35: {{2},{1,1}}
%t A326533 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326533 Select[Range[100],UnsameQ@@PrimeOmega/@primeMS[#]&]
%Y A326533 A subsequence of A005117.
%Y A326533 Cf. A007837, A038041, A112798, A302242, A320324, A326026, A326514, A326517, A326534, A326535, A326536, A326537.
%K A326533 nonn
%O A326533 1,2
%A A326533 _Gus Wiseman_, Jul 12 2019