cp's OEIS Frontend

A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

%I A386587 #7 Aug 06 2025 22:39:07
%S A386587 1,1,1,1,1,0,1,2,1,0,1,1,1,0,0,2,1,1,1,1,0,0,1,1,1,0,2,1,1,0,1,3,0,0,
%T A386587 0,0,1,0,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,1,4,0,0,1,1,
%U A386587 0,0,1,1,1,0,1,1,0,0,1,1,2,0,1,0,0,0,0
%N A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.
%C A386587 First differs from A382525 at a(216) = 1, A382525(216) = 2.
%e A386587 The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.
%t A386587 disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
%t A386587 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A386587 Table[Length[disjointFamilies[prix[n]]],{n,100}]
%Y A386587 Positions of positive terms are A351294, conjugate A381432.
%Y A386587 Positions of 0 are A351295, conjugate A381433.
%Y A386587 For ordered set partitions we have A382525.
%Y A386587 Positions of first appearances are A382775.
%Y A386587 The separable case is A386575.
%Y A386587 The inseparable case is A386582, see A386632.
%Y A386587 A000110 counts set partitions, ordered A000670.
%Y A386587 A003242 and A335452 count separations, ranks A333489.
%Y A386587 A239455 counts Look-and-Say partitions, complement A351293.
%Y A386587 A279790 counts disjoint families on strongly normal multisets.
%Y A386587 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386587 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386587 A386633 counts separable set partitions, row sums of A386635.
%Y A386587 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386587 Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.
%K A386587 nonn
%O A386587 1,8
%A A386587 _Gus Wiseman_, Aug 06 2025