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A382304 MM-numbers of multiset partitions into sets with a common sum.

%I A382304 #8 Apr 03 2025 14:57:57
%S A382304 1,2,3,4,5,8,9,11,13,16,17,25,27,29,31,32,41,43,47,59,64,67,73,79,81,
%T A382304 83,101,109,113,121,125,127,128,137,139,143,149,157,163,167,169,179,
%U A382304 181,191,199,211,233,241,243,256,257,269,271,277,283,289,293,313,317
%N A382304 MM-numbers of multiset partitions into sets with a common sum.
%C A382304 Also products of prime numbers of squarefree index with a common sum of prime indices.
%C A382304 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, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%H A382304 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%F A382304 Equals A302478 /\ A326534.
%e A382304 The terms together with their prime indices of prime indices begin:
%e A382304    1: {}
%e A382304    2: {{}}
%e A382304    3: {{1}}
%e A382304    4: {{},{}}
%e A382304    5: {{2}}
%e A382304    8: {{},{},{}}
%e A382304    9: {{1},{1}}
%e A382304   11: {{3}}
%e A382304   13: {{1,2}}
%e A382304   16: {{},{},{},{}}
%e A382304   17: {{4}}
%e A382304   25: {{2},{2}}
%e A382304   27: {{1},{1},{1}}
%e A382304   29: {{1,3}}
%e A382304   31: {{5}}
%e A382304   32: {{},{},{},{},{}}
%t A382304 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A382304 Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@UnsameQ@@@prix/@prix[#]&]
%Y A382304 Set partitions of this type are counted by A035470.
%Y A382304 Twice-partitions of this type are counted by A279788.
%Y A382304 For just strict blocks we have A302478.
%Y A382304 For just a common sum we have A326534, distinct sums A326535.
%Y A382304 Factorizations of this type are counted by A382080.
%Y A382304 For distinct instead of equal sums we have A382201.
%Y A382304 For constant instead of strict blocks we have A382215.
%Y A382304 Normal multiset partitions of this type are counted by A382429.
%Y A382304 A055396 gives least prime index, greatest A061395.
%Y A382304 A056239 adds up prime indices, row sums of A112798.
%Y A382304 A058891 counts set-systems, covering A003465, connected A323818.
%Y A382304 Cf. A000720, A005117, A050320, A050326, A293511, A302242, A302494, A381635, A381636, A381995.
%K A382304 nonn
%O A382304 1,2
%A A382304 _Gus Wiseman_, Apr 01 2025