cp's OEIS Frontend

A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

%I A381452 #11 Mar 08 2025 12:24:03
%S A381452 1,1,1,1,1,2,1,2,1,2,1,3,1,2,2,2,1,3,1,3,2,2,1,4,1,2,2,3,1,5,1,3,2,2,
%T A381452 2,4,1,2,2,5,1,5,1,3,3,2,1,5,1,3,2,3,1,5,2,5,2,2,1,7,1,2,3,4,2,5,1,3,
%U A381452 2,5,1,6,1,2,3,3,2,5,1,6,2,2,1,8,2,2,2
%N A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.
%C A381452 First differs from A045778 at a(24) = 4, A045778(24) = 5.
%C A381452 Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into distinct factors > 1.
%C A381452 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 A381452 A multiset partition can be regarded as an arrow in the 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 A381452 Sets of multisets are generally not transitive. For example, we have arrows: {{1},{2},{1,2}}: {1,1,2,2} -> {1,2,3} and {{1,2},{3}}: {1,2,3} -> {3,3}, but there is no set of multisets {1,1,2,2} -> {3,3}.
%F A381452 a(A002110(n)) = A066723(n).
%e A381452 The prime indices of 24 are {1,1,1,2}, with 5 partitions into a set of multisets:
%e A381452   {{1,1,1,2}}
%e A381452   {{1},{1,1,2}}
%e A381452   {{2},{1,1,1}}
%e A381452   {{1,1},{1,2}}
%e A381452   {{1},{2},{1,1}}
%e A381452 with block-sums: {5}, {1,4}, {2,3}, {2,3}, {1,2,2}, of which 4 are distinct, so a(24) = 4.
%t A381452 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381452 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381452 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A381452 Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&]]],{n,100}]
%Y A381452 Before taking sums we had A045778.
%Y A381452 If each block is a set we have A381441, before sums A050326.
%Y A381452 For distinct block-sums instead of blocks we have A381637, before sums A321469.
%Y A381452 Other multiset partitions of prime indices:
%Y A381452 - For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
%Y A381452 - For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
%Y A381452 - For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
%Y A381452 - For sets of constant multisets (A050361) see A381715.
%Y A381452 - For set systems with distinct sums (A381633) see A381634, zeros A293243.
%Y A381452 - For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
%Y A381452 More on sets of multisets: A261049, A317776, A317775, A296118, A318286.
%Y A381452 A000041 counts integer partitions, strict A000009.
%Y A381452 A000040 lists the primes.
%Y A381452 A003963 gives product of prime indices.
%Y A381452 A055396 gives least prime index, greatest A061395.
%Y A381452 A056239 adds up prime indices, row sums of A112798.
%Y A381452 A122111 represents conjugation in terms of Heinz numbers.
%Y A381452 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381452 Cf. A000720, A001222, A001970, A002846, A066328, A213385, A213427, A299200, A299202, A300385, A317142.
%K A381452 nonn
%O A381452 1,6
%A A381452 _Gus Wiseman_, Mar 06 2025