cp's OEIS Frontend

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

%I A387110 #15 Aug 21 2025 10:04:06
%S A387110 1,1,2,0,3,2,5,0,2,3,7,0,11,5,6,0,15,2,22,0,10,7,30,0,6,11,0,0,42,6,
%T A387110 56,0,14,15,15,0,77,22,22,0,101,10,135,0,6,30,176,0,20,6,30,0,231,0,
%U A387110 21,0,44,42,297,0,385,56,10,0,33,14,490,0,60,15,627,0
%N A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.
%C A387110 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 A387110 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 A387110 The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
%e A387110   ((2),(1,1))
%e A387110   ((1,1),(2))
%e A387110 The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
%e A387110   ((2),(3))
%e A387110   ((2),(2,1))
%e A387110   ((2),(1,1,1))
%e A387110   ((1,1),(2,1))
%e A387110   ((1,1),(1,1,1))
%t A387110 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387110 Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]
%Y A387110 Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
%Y A387110 Allowing repeated partitions gives A299200, A357977, A357982, A357978.
%Y A387110 For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
%Y A387110 For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
%Y A387110 For prime factors instead of partitions see A355741, A355742, A387136.
%Y A387110 The disjoint case is A383706.
%Y A387110 For initial intervals instead of partitions we have A387111.
%Y A387110 The case of strict partitions is A387115.
%Y A387110 The case of constant partitions is A387120.
%Y A387110 Taking each prime factor (instead of index) gives A387133.
%Y A387110 A000041 counts integer partitions, strict A000009.
%Y A387110 A003963 multiplies together prime indices.
%Y A387110 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387110 A120383 lists numbers divisible by all of their prime indices.
%Y A387110 A289509 lists numbers with relatively prime prime indices.
%Y A387110 Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.
%K A387110 nonn,new
%O A387110 1,3
%A A387110 _Gus Wiseman_, Aug 18 2025