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A326535 MM-numbers of multiset partitions where each part has a different sum.

%I A326535 #10 Apr 03 2025 02:20:19
%S A326535 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,37,38,39,
%T A326535 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,71,73,74,77,78,79,
%U A326535 82,83,85,86,87,89,91,93,94,95,97,101,102,103,106,107,109
%N A326535 MM-numbers of multiset partitions where each part has a different sum.
%C A326535 First differs from A298540 in having 187.
%C A326535 These are numbers where each prime index has a different sum 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 A326535 Robert Price, <a href="/A326535/b326535.txt">Table of n, a(n) for n = 1..9715</a>
%H A326535 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 A326535 The sequence of multiset partitions where each part has a different sum, preceded by their MM-numbers, begins:
%e A326535    1: {}
%e A326535    2: {{}}
%e A326535    3: {{1}}
%e A326535    5: {{2}}
%e A326535    6: {{},{1}}
%e A326535    7: {{1,1}}
%e A326535   10: {{},{2}}
%e A326535   11: {{3}}
%e A326535   13: {{1,2}}
%e A326535   14: {{},{1,1}}
%e A326535   15: {{1},{2}}
%e A326535   17: {{4}}
%e A326535   19: {{1,1,1}}
%e A326535   21: {{1},{1,1}}
%e A326535   22: {{},{3}}
%e A326535   23: {{2,2}}
%e A326535   26: {{},{1,2}}
%e A326535   29: {{1,3}}
%e A326535   30: {{},{1},{2}}
%e A326535   31: {{5}}
%t A326535 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326535 Select[Range[100],UnsameQ@@Total/@primeMS/@primeMS[#]&]
%Y A326535 Cf. A038041, A112798, A275780, A302242, A320324, A321469, A326519, A326533, A326534, A326536, A326537.
%K A326535 nonn
%O A326535 1,2
%A A326535 _Gus Wiseman_, Jul 12 2019