A326533 - OEIS

%I #8 Jul 13 2019 09:12:46

%S 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 43,46,47,53,57,58,59,61,62,65,67,69,70,71,73,74,77,78,79,82,83,86,87,

%U 89,94,95,97,101,103,106,107,109,111,113,114,115,118,119,122

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

%C 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 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 The sequence of multiset partitions where each part has a different average preceded by their MM-numbers begins:

%e    1: {}

%e    2: {{}}

%e    3: {{1}}

%e    5: {{2}}

%e    6: {{},{1}}

%e    7: {{1,1}}

%e   10: {{},{2}}

%e   11: {{3}}

%e   13: {{1,2}}

%e   14: {{},{1,1}}

%e   17: {{4}}

%e   19: {{1,1,1}}

%e   21: {{1},{1,1}}

%e   22: {{},{3}}

%e   23: {{2,2}}

%e   26: {{},{1,2}}

%e   29: {{1,3}}

%e   31: {{5}}

%e   34: {{},{4}}

%e   35: {{2},{1,1}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],UnsameQ@@PrimeOmega/@primeMS[#]&]

%Y A subsequence of A005117.

%Y Cf. A007837, A038041, A112798, A302242, A320324, A326026, A326514, A326517, A326534, A326535, A326536, A326537.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 12 2019