cp's OEIS Frontend

A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

%I A381441 #10 Mar 08 2025 12:23:37
%S A381441 1,1,1,0,1,2,1,0,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,0,2,0,1,1,5,1,0,2,2,
%T A381441 2,1,1,2,2,0,1,5,1,1,1,2,1,0,0,1,2,1,1,0,2,0,2,2,1,4,1,2,1,0,2,5,1,1,
%U A381441 2,5,1,0,1,2,1,1,2,5,1,0,0,2,1,4,2,2,2
%N A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.
%C A381441 First differs from A050326 at a(210) = 13, A050326(210) = 15. This comes from the set systems {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
%C A381441 Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a strict factorization of n into squarefree numbers > 1.
%C A381441 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.
%C A381441 A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
%C A381441 Sets of sets are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set of sets {1,1,2} -> {4}.
%F A381441 a(A002110(n)) = A066723(n).
%e A381441 The prime indices of 60 are {1,1,2,3}, with partitions into sets of sets:
%e A381441   {{1},{1,2,3}}
%e A381441   {{1,2},{1,3}}
%e A381441   {{1},{2},{1,3}}
%e A381441   {{1},{3},{1,2}}
%e A381441 with block-sums: {1,6}, {3,4}, {1,2,4}, {1,3,3}, which are all different, so a(60) = 4.
%t A381441 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381441 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A381441 Table[Length[Union[Sort[Total/@prix/@#]&/@Select[facs[n],UnsameQ@@#&&And@@SquareFreeQ/@#&]]],{n,100}]
%Y A381441 Before taking sums we had A050326, non-strict A050320.
%Y A381441 Positions of 0 are A293243.
%Y A381441 Positions of 1 are A293511.
%Y A381441 This is the strict version of A381078 (lower A381454).
%Y A381441 For distinct block-sums (instead of blocks) we have A381634, before sums A381633.
%Y A381441 Other multiset partitions of prime indices:
%Y A381441 - For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
%Y A381441 - For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
%Y A381441 - For strict multiset partitions (A045778) see A381452.
%Y A381441 - For sets of constant multisets (A050361) see A381715.
%Y A381441 - For strict multiset partitions with distinct sums (A321469) see A381637.
%Y A381441 - For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
%Y A381441 More on set systems: A050342, A116539, A279785, A296120, A318361.
%Y A381441 A000041 counts integer partitions, strict A000009.
%Y A381441 A000040 lists the primes, differences A001223.
%Y A381441 A003963 gives product of prime indices.
%Y A381441 A055396 gives least prime index, greatest A061395.
%Y A381441 A056239 adds up prime indices, row sums of A112798.
%Y A381441 A122111 represents conjugation in terms of Heinz numbers.
%Y A381441 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381441 Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A299202, A300385, A317142.
%K A381441 nonn
%O A381441 1,6
%A A381441 _Gus Wiseman_, Mar 06 2025