cp's OEIS Frontend

A326534 MM-numbers of multiset partitions where every part has the same sum.

%I A326534 #7 Jul 13 2019 09:12:55
%S A326534 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 A326534 53,59,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,
%U A326534 128,131,137,139,143,149,151,157,163,167,169,173,175,179,181,191
%N A326534 MM-numbers of multiset partitions where every part has the same sum.
%C A326534 First differs from A298538 in lacking 187.
%C A326534 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 A326534 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 A326534 The sequence of multiset partitions where every part has the same sum, preceded by their MM-numbers, begins:
%e A326534    1: {}
%e A326534    2: {{}}
%e A326534    3: {{1}}
%e A326534    4: {{},{}}
%e A326534    5: {{2}}
%e A326534    7: {{1,1}}
%e A326534    8: {{},{},{}}
%e A326534    9: {{1},{1}}
%e A326534   11: {{3}}
%e A326534   13: {{1,2}}
%e A326534   16: {{},{},{},{}}
%e A326534   17: {{4}}
%e A326534   19: {{1,1,1}}
%e A326534   23: {{2,2}}
%e A326534   25: {{2},{2}}
%e A326534   27: {{1},{1},{1}}
%e A326534   29: {{1,3}}
%e A326534   31: {{5}}
%e A326534   32: {{},{},{},{},{}}
%e A326534   35: {{2},{1,1}}
%t A326534 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326534 Select[Range[100],SameQ@@Total/@primeMS/@primeMS[#]&]
%Y A326534 Cf. A035470, A038041, A112798, A302242, A320324, A321455, A326518, A326533, A326535, A326536, A326537.
%K A326534 nonn
%O A326534 1,2
%A A326534 _Gus Wiseman_, Jul 12 2019