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

1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 145, 146, 149, 155, 157, 158, 163, 165

Offset: 1

Author

Gus Wiseman, Mar 21 2025

Keywords

• nonn

Comments

First differs from A302494 in lacking 143, corresponding to the multiset partition {{1,2},{3}}.

Also products of prime numbers of squarefree index such that the factors all have distinct sums 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, 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}}.

Examples

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  22: {{},{3}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  39: {{1},{1,2}}
		

Crossrefs

Set partitions of this type are counted by A275780.

Twice-partitions of this type are counted by A279785.

For just sets of sets we have A302478.

For distinct blocks instead of block-sums we have A302494.

For equal instead of distinct sums we have A302497.

For just distinct sums we have A326535.

For normal multiset partitions see A326519, A326533, A326537, A381718.

Factorizations of this type are counted by A381633. See also A001055, A045778, A050320, A050326, A321455, A321469, A382080.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

Cf. A000720, A003963, A005117, A007716, A293511, A302242, A319899, A326534, A368100, A368101, A381635, A382215.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@SquareFreeQ/@prix[#]&&UnsameQ@@Total/@prix/@prix[#]&]

Formula

Equals A302478 /\ A326535.