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

This is the strict version of A355731, firsts A355732, prime factors A355741, prime powers A355742.

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

A289509 lists numbers with relatively prime prime indices.

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

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

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

For weakly increasing instead of strict we have A355735, prime factors A355745.

Cf. A000720, A076610, `~A302796, `A355535, `A355537, A355733, `A355735, A355744, `A355747, `A355748.

allocated for Gus WisemanNumber of ways to choose a sequence of all different divisors of each prime index of n.

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

1,3

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.

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

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

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

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

This is the strict version of A355731, firsts A355732, prime factors A355741, prime powers A355742.

For weakly increasing instead of strict we have A355735, prime factors A355745.

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

Positions of 0's are A355740.

A000005 counts divisors.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

A120383 lists numbers divisible by all of their prime indices.

A289508 gives GCD of prime indices.

A289509 lists numbers with relatively prime prime indices.

Cf. A000720, A076610, `~A302796, `A355535, `A355537, A355733, `A355744, `A355747, `A355748.

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Gus Wiseman, Jul 18 2022

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editing

allocated for Gus Wiseman

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