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A382201 MM-numbers of sets of sets with distinct sums.

%I A382201 #8 Mar 23 2025 08:40:28
%S A382201 1,2,3,5,6,10,11,13,15,17,22,26,29,30,31,33,34,39,41,43,47,51,55,58,
%T A382201 59,62,65,66,67,73,78,79,82,83,85,86,87,93,94,101,102,109,110,113,118,
%U A382201 123,127,129,130,134,137,139,141,145,146,149,155,157,158,163,165
%N A382201 MM-numbers of sets of sets with distinct sums.
%C A382201 First differs from A302494 in lacking 143, corresponding to the multiset partition {{1,2},{3}}.
%C A382201 Also products of prime numbers of squarefree index such that the factors all have distinct sums of prime indices.
%C A382201 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}}.
%F A382201 Equals A302478 /\ A326535.
%e A382201 The terms together with their prime indices of prime indices begin:
%e A382201    1: {}
%e A382201    2: {{}}
%e A382201    3: {{1}}
%e A382201    5: {{2}}
%e A382201    6: {{},{1}}
%e A382201   10: {{},{2}}
%e A382201   11: {{3}}
%e A382201   13: {{1,2}}
%e A382201   15: {{1},{2}}
%e A382201   17: {{4}}
%e A382201   22: {{},{3}}
%e A382201   26: {{},{1,2}}
%e A382201   29: {{1,3}}
%e A382201   30: {{},{1},{2}}
%e A382201   31: {{5}}
%e A382201   33: {{1},{3}}
%e A382201   34: {{},{4}}
%e A382201   39: {{1},{1,2}}
%t A382201 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A382201 Select[Range[100],And@@SquareFreeQ/@prix[#]&&UnsameQ@@Total/@prix/@prix[#]&]
%Y A382201 Set partitions of this type are counted by A275780.
%Y A382201 Twice-partitions of this type are counted by A279785.
%Y A382201 For just sets of sets we have A302478.
%Y A382201 For distinct blocks instead of block-sums we have A302494.
%Y A382201 For equal instead of distinct sums we have A302497.
%Y A382201 For just distinct sums we have A326535.
%Y A382201 For normal multiset partitions see A326519, A326533, A326537, A381718.
%Y A382201 Factorizations of this type are counted by A381633. See also A001055, A045778, A050320, A050326, A321455, A321469, A382080.
%Y A382201 A055396 gives least prime index, greatest A061395.
%Y A382201 A056239 adds up prime indices, row sums of A112798.
%Y A382201 Cf. A000720, A003963, A005117, A007716, A293511, A302242, A319899, A326534, A368100, A368101, A381635, A382215.
%K A382201 nonn
%O A382201 1,2
%A A382201 _Gus Wiseman_, Mar 21 2025