cp's OEIS Frontend

A387120 Number of ways to choose a different constant integer partition of each prime index of n.

%I A387120 #6 Aug 28 2025 17:25:09
%S A387120 1,1,2,0,2,2,3,0,2,2,2,0,4,3,4,0,2,2,4,0,6,2,3,0,2,4,0,0,4,4,2,0,4,2,
%T A387120 6,0,6,4,8,0,2,6,4,0,4,3,4,0,6,2,4,0,5,0,4,0,8,4,2,0,6,2,6,0,8,4,2,0,
%U A387120 6,6,6,0,4,6,4,0,6,8,4,0,0,2,2,0,4,4,8
%N A387120 Number of ways to choose a different constant integer partition of each prime index of n.
%C A387120 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.
%e A387120 The prime indices of 90 are {1,2,2,3}, with choices:
%e A387120   ((1),(2),(1,1),(3))
%e A387120   ((1),(1,1),(2),(3))
%e A387120   ((1),(2),(1,1),(1,1,1))
%e A387120   ((1),(1,1),(2),(1,1,1))
%e A387120 so a(90) = 4.
%t A387120 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387120 Table[Length[Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]],UnsameQ@@#&]],{n,100}]
%Y A387120 For multiset systems see A355529, set systems A367901.
%Y A387120 For not necessarily different choices we have A355731, see A355740.
%Y A387120 For divisors instead of constant partitions we have A355739 (also the disjoint case).
%Y A387120 For prime factors instead of constant partitions we have A387136.
%Y A387120 For all instead of just constant partitions we have A387110, disjoint case A383706.
%Y A387120 For initial intervals instead of partitions we have A387111.
%Y A387120 For strict instead of constant partitions we have A387115.
%Y A387120 Twice partitions of this type are counted by A387179, constant-block case of A296122.
%Y A387120 Positions of zero are A387180 (non-choosable), complement A387181 (choosable).
%Y A387120 A000041 counts integer partitions, strict A000009.
%Y A387120 A003963 multiplies together prime indices.
%Y A387120 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387120 A120383 lists numbers divisible by all of their prime indices.
%Y A387120 A289509 lists numbers with relatively prime prime indices.
%Y A387120 Cf. A000005, A063834, A261049, A299200, A299201, A335433, A335448, A355741, A357977, A387135.
%K A387120 nonn,new
%O A387120 1,3
%A A387120 _Gus Wiseman_, Aug 26 2025