cp's OEIS Frontend

A381633 Number of ways to partition the prime indices of n into sets with distinct sums.

%I A381633 #6 Mar 09 2025 20:26:09
%S A381633 1,1,1,0,1,2,1,0,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,0,2,0,1,1,4,1,0,2,2,
%T A381633 2,1,1,2,2,0,1,5,1,1,1,2,1,0,0,1,2,1,1,0,2,0,2,2,1,3,1,2,1,0,2,5,1,1,
%U A381633 2,4,1,0,1,2,1,1,2,5,1,0,0,2,1,4,2,2,2
%N A381633 Number of ways to partition the prime indices of n into sets with distinct sums.
%C A381633 First differs from A050326 at 30, 60, 70, 90, ...
%C A381633 First differs from A339742 at 42, 66, 78, 84, ...
%C A381633 First differs from A381634 at a(210) = 12, A381634(210) = 10.
%C A381633 Also the number of factorizations on n into squarefree numbers > 1 with distinct sums of prime indices.
%C A381633 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 A381633 The A050320(60) = 6 ways to partition {1,1,2,3} into sets are:
%e A381633   {{1},{1,2,3}}
%e A381633   {{1,2},{1,3}}
%e A381633   {{1},{1},{2,3}}
%e A381633   {{1},{2},{1,3}}
%e A381633   {{1},{3},{1,2}}
%e A381633   {{1},{1},{2},{3}}
%e A381633 Of these, only the following have distinct block-sums:
%e A381633   {{1},{1,2,3}}
%e A381633   {{1,2},{1,3}}
%e A381633   {{1},{2},{1,3}}
%e A381633 So a(60) = 3.
%t A381633 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381633 sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A381633 Table[Length[Select[sfacs[n],UnsameQ@@hwt/@#&]],{n,100}]
%Y A381633 Without distinct block-sums we have A050320, after sums A381078 (lower A381454).
%Y A381633 For distinct blocks instead of sums we have A050326, after sums A381441, see A358914.
%Y A381633 Taking block-sums (and sorting) gives A381634.
%Y A381633 For constant instead of strict blocks we have A381635, see A381716, A381636.
%Y A381633 Positions of 0 are A381806, superset of A293243.
%Y A381633 Positions of 1 are A381870, superset of A293511.
%Y A381633 More on set multipartitions with distinct sums: A279785, A381717, A381718.
%Y A381633 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381633 A000041 counts integer partitions, strict A000009.
%Y A381633 A001055 count multiset partitions of prime indices, see A317141 (upper), A300383 (lower).
%Y A381633 A003963 gives product of prime indices.
%Y A381633 A055396 gives least prime index, greatest A061395.
%Y A381633 A056239 adds up prime indices, row sums of A112798.
%Y A381633 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381633 Cf. A000720, A001222, A002846, A005117, A213242, A299202, A300385, A317142, A317143, A321469.
%K A381633 nonn
%O A381633 1,6
%A A381633 _Gus Wiseman_, Mar 09 2025