%I #7 Jul 19 2022 08:04:21

%S 1,1,2,0,2,1,3,0,2,1,2,0,4,2,3,0,2,0,4,0,4,1,3,0,2,3,0,0,4,1,2,0,3,1,

%T 5,0,6,3,6,0,2,1,4,0,2,2,4,0,6,0,3,0,5,0,3,0,6,3,2,0,6,1,2,0,6,1,2,0,

%U 5,2,6,0,4,5,2,0,5,2,4,0,0,1,2,0,3,3,6

%N Number of ways to choose a sequence of all different divisors, one of each prime index of n (with multiplicity).

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cartesian_product">Cartesian product</a>.

%e The a(49) = 6 ways are: (1,2), (1,4), (2,1), (2,4), (4,1), (4,2).

%e The a(182) = 5 ways are: (1,2,3), (1,2,6), (1,4,2), (1,4,3), (1,4,6).

%e The a(546) = 2 ways are: (1,2,4,3), (1,2,4,6).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Length[Select[Tuples[Divisors/@primeMS[n]],UnsameQ@@#&]],{n,100}]

%Y This is the strict version of A355731, firsts A355732.

%Y For relatively prime instead of strict we have A355737, firsts A355738.

%Y Positions of 0's are A355740.

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors, with sum A001414.

%Y A001222 counts prime factors with multiplicity.

%Y A003963 multiplies together the prime indices of n.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A120383 lists numbers divisible by all of their prime indices.

%Y A289508 gives GCD of prime indices, positions of 1's A289509.

%Y Cf. A000720, A076610, A302796, A355535, A355537, A355733, A355735, A355741, A355742, A355744, A355748.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 18 2022