cp's OEIS Frontend

A387136 Number of ways to choose a sequence of distinct prime factors, one of each prime index of 2n - 1.

%I A387136 #7 Sep 01 2025 12:23:46
%S A387136 1,1,1,1,0,1,2,1,1,1,0,1,0,0,2,1,1,1,2,1,1,2,0,2,0,1,1,1,0,1,2,0,1,1,
%T A387136 1,2,2,0,1,2,0,1,1,1,2,1,1,1,1,0,2,1,0,2,1,1,3,0,0,1,0,1,0,1,1,1,0,0,
%U A387136 2,2,2,2,2,0,2,2,0,1,1,0,1,2,1,2,2,0,2
%N A387136 Number of ways to choose a sequence of distinct prime factors, one of each prime index of 2n - 1.
%C A387136 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 A387136 The prime indices of 4537 are {6,70}, with choices (2,5), (2,7), (3,2), (3,5), (3,7). Since 4537 = 2 * 2269 - 1, we have a(2269) = 5.
%t A387136 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387136 Table[Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@prix[2n-1]],UnsameQ@@#&]],{n,100}]
%Y A387136 Here we use the version with alternating zeros (put n instead of 2n - 1 in the name).
%Y A387136 Twice partitions of this type are counted by A296122.
%Y A387136 Positions of zero are A355529, complement A368100.
%Y A387136 For divisors instead of prime factors we have A355739.
%Y A387136 Allowing repeated choices gives A355741.
%Y A387136 For partitions instead of prime factors we have A387110.
%Y A387136 For initial intervals instead of prime factors we have A387111.
%Y A387136 For strict partitions instead of prime factors we have A387115, disjoint case A383706.
%Y A387136 For constant partitions instead of prime factors we have A387120.
%Y A387136 A000041 counts integer partitions, strict A000009.
%Y A387136 A003963 multiplies together prime indices.
%Y A387136 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387136 A120383 lists numbers divisible by all of their prime indices.
%Y A387136 A289509 lists numbers with relatively prime prime indices.
%Y A387136 Cf. A261049, A276078, A276079, A299200, A335433, A335448, A355731, A355745, A367771, A387133, A387135, A387327.
%K A387136 nonn,new
%O A387136 1,7
%A A387136 _Gus Wiseman_, Aug 30 2025