cp's OEIS Frontend

A387115 Number of ways to choose a sequence of distinct strict integer partitions, one of each prime index of n.

%I A387115 #9 Aug 21 2025 10:03:54
%S A387115 1,1,1,0,2,1,2,0,0,2,3,0,4,2,2,0,5,0,6,0,2,3,8,0,2,4,0,0,10,2,12,0,3,
%T A387115 5,4,0,15,6,4,0,18,2,22,0,0,8,27,0,2,2,5,0,32,0,6,0,6,10,38,0,46,12,0,
%U A387115 0,8,3,54,0,8,4,64,0,76,15,2,0,6,4,89,0,0
%N A387115 Number of ways to choose a sequence of distinct strict integer partitions, one of each prime index of n.
%C A387115 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 A387115 The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.
%e A387115 The prime indices of 15 are (2,3), and there are a(15) = 2 choices:
%e A387115   ((2),(3))
%e A387115   ((2),(2,1))
%e A387115 The prime indices of 121 are (5,5), and there are a(121) = 6 choices:
%e A387115   ((5),(4,1))
%e A387115   ((5),(3,2))
%e A387115   ((4,1),(5))
%e A387115   ((4,1),(3,2))
%e A387115   ((3,2),(5))
%e A387115   ((3,2),(4,1))
%t A387115 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387115 Table[Length[Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[n]],UnsameQ@@#&]],{n,100}]
%Y A387115 For divisors instead of partitions we have A355739, see A355740, A355733, A355734.
%Y A387115 Allowing repeated partitions gives A357982, see A299200, A357977, A357978.
%Y A387115 Twice-partitions of this type are counted by A358914, strict case of A270995.
%Y A387115 The disjoint case is A383706.
%Y A387115 Allowing non-strict partitions gives A387110, for prime factors A387133.
%Y A387115 For initial intervals instead of strict partitions we have A387111.
%Y A387115 For constant instead of strict partitions we have A387120.
%Y A387115 Positions of 0 are A387176 (non-choosable), complement A387177 (choosable).
%Y A387115 A000041 counts integer partitions, strict A000009.
%Y A387115 A003963 multiplies together the prime indices of n.
%Y A387115 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387115 A120383 lists numbers divisible by all of their prime indices.
%Y A387115 A289509 lists numbers with relatively prime prime indices.
%Y A387115 Cf. A000720, A063834, A261049, A335433, A335448, A355731, A355741, A355742, A355744, A357980.
%K A387115 nonn,easy,new
%O A387115 1,5
%A A387115 _Gus Wiseman_, Aug 20 2025