%I #7 Jul 13 2019 09:12:55

%S 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,35,37,41,43,47,49,

%T 53,59,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,

%U 128,131,137,139,143,149,151,157,163,167,169,173,175,179,181,191

%N MM-numbers of multiset partitions where every part has the same sum.

%C First differs from A298538 in lacking 187.

%C These are numbers where each prime index has the same 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 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 every part has the same sum, preceded by their MM-numbers, begins:

%e 1: {}

%e 2: {{}}

%e 3: {{1}}

%e 4: {{},{}}

%e 5: {{2}}

%e 7: {{1,1}}

%e 8: {{},{},{}}

%e 9: {{1},{1}}

%e 11: {{3}}

%e 13: {{1,2}}

%e 16: {{},{},{},{}}

%e 17: {{4}}

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

%e 23: {{2,2}}

%e 25: {{2},{2}}

%e 27: {{1},{1},{1}}

%e 29: {{1,3}}

%e 31: {{5}}

%e 32: {{},{},{},{},{}}

%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],SameQ@@Total/@primeMS/@primeMS[#]&]

%Y Cf. A035470, A038041, A112798, A302242, A320324, A321455, A326518, A326533, A326535, A326536, A326537.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 12 2019