cp's OEIS Frontend

A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.

%I A384389 #13 Jun 03 2025 08:43:51
%S A384389 0,0,0,0,1,0,1,0,0,0,2,0,3,0,0,0,4,0,5,0,1,1,7,0,2,1,0,0,9,0,11,0,1,2,
%T A384389 1,0,14,2,1,0,17,0,21,0,0,4,26,0,2,0,2,0,31,0,2,0,3,4,37,0,45,6,0,0,3,
%U A384389 0,53,0,4,0,63,0,75,7,0,0,2,0,88,0,0,9
%N A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.
%C A384389 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 A384389 By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.
%F A384389 a(prime(n)) = A000009(n) - 1.
%e A384389 The prime indices of 65 are {3,6}, and we have proper choices: ((3),(5,1)), ((3),(4,2)), ((2,1),(6)). Hence a(65) = 3.
%e A384389 The prime indices of 175 are {3,3,4}, and we have choices: ((3),(2,1),(4)), ((2,1),(3),(4)), both already proper. Hence a(175) = 2.
%t A384389 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384389 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384389 Table[Length[pofprop[prix[n]]],{n,100}]
%Y A384389 Without disjointness we have A357982 - 1, non-strict version A299200 - 1.
%Y A384389 This is the proper case of A383706, conjugate version A384005.
%Y A384389 Positions of positive terms are A384321.
%Y A384389 Positions of 0 are A384349.
%Y A384389 Positions of 1 are A384390.
%Y A384389 Positions of terms > 1 are A384393.
%Y A384389 The conjugate version is A384394.
%Y A384389 Positions of first appearances are A384396.
%Y A384389 A000041 counts integer partitions, strict A000009.
%Y A384389 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384389 A055396 gives least prime index, greatest A061395.
%Y A384389 A056239 adds up prime indices, row sums of A112798.
%Y A384389 A239455 counts Look-and-Say partitions, ranks A351294
%Y A384389 A351293 counts non-Look-and-Say partitions, ranks A351295.
%Y A384389 Cf. A382912, A383710, A383711, A382913, A383708, A383533.
%Y A384389 Cf. A179009, A279375, A279790, A381454, A382525, A384322, A384347.
%K A384389 nonn
%O A384389 1,11
%A A384389 _Gus Wiseman_, Jun 01 2025