cp's OEIS Frontend

A381872 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks having a common sum.

%I A381872 #6 Mar 14 2025 17:10:14
%S A381872 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,3,1,1,1,1,1,1,1,1,2,1,2,1,1,2,1,2,1,1,
%T A381872 1,3,1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,1,1,1,1,
%U A381872 1,2,1,1,1,1,1,1,1,1,1,1,3,1,1,2,1,1,1
%N A381872 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks having a common sum.
%C A381872 First differs from A321455 at a(144) = 4, A321455(144) = 3.
%C A381872 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.
%e A381872 The prime indices of 144 are {1,1,1,1,2,2}, with the following 4 multiset partitions having common block sum:
%e A381872   {{1,1,1,1,2,2}}
%e A381872   {{2,2},{1,1,1,1}}
%e A381872   {{1,1,2},{1,1,2}}
%e A381872   {{2},{2},{1,1},{1,1}}
%e A381872 with sums: 8, 4, 4, 2, of which 3 are distinct, so a(144) = 3.
%e A381872 The prime indices of 1296 are {1,1,1,1,2,2,2,2}, with the following 7 multiset partitions having common block sum:
%e A381872   {{1,1,1,1,2,2,2,2}}
%e A381872   {{2,2,2},{1,1,1,1,2}}
%e A381872   {{1,1,2,2},{1,1,2,2}}
%e A381872   {{2,2},{2,2},{1,1,1,1}}
%e A381872   {{2,2},{1,1,2},{1,1,2}}
%e A381872   {{1,2},{1,2},{1,2},{1,2}}
%e A381872   {{2},{2},{2},{2},{1,1},{1,1}}
%e A381872 with sums: 12, 6, 6, 4, 4, 3, 2, of which 5 are distinct, so a(1296) = 5.
%t A381872 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381872 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381872 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A381872 Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],SameQ@@Total/@#&]]],{n,100}]
%Y A381872 With equal blocks instead of sums we have A089723.
%Y A381872 Without equal sums we have A317141, before sums A001055, lower A300383.
%Y A381872 Positions of terms > 1 are A321454.
%Y A381872 Before taking sums we had A321455.
%Y A381872 With distinct instead of equal sums we have A381637, before sums A321469.
%Y A381872 A000041 counts integer partitions, strict A000009, constant A000005.
%Y A381872 A055396 gives least prime index, greatest A061395.
%Y A381872 A056239 adds up prime indices, row sums of A112798.
%Y A381872 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381872 Other multiset partitions of prime indices:
%Y A381872 - For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
%Y A381872 - For sets of constant multisets (A050361) see A381715.
%Y A381872 - For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
%Y A381872 Cf. A000720, A000961, A001222, A279787, A279789, A305551, A306017, A321451, A321452, A321453.
%K A381872 nonn
%O A381872 1,4
%A A381872 _Gus Wiseman_, Mar 14 2025