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The prime indices of 1469 are {6,30}, and there are 5 five valid choices: (2,2), (2,3), (2,5), (3,3), (3,5), so a(1469) = 5.

The a(prime indices of 1469) = 5 ways are {6,30}, and there are 5 valid choices: {(2,2}, {), (2,3}, {), (2,5}, {), (3,3}, {), (3,5}), so a(1469) = 5.

Choosing a multiset instead of weakly increasing sequence gives A355744.

Cf. A000720, A076610, `A335433, ~`A335448, `~A340852, A355737, A355739, `A355740, A355742.

allocated for Gus WisemanNumber of ways to choose a prime factor of each prime index of n (with multiplicity, in weakly increasing order) such that the result is also weakly increasing.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

1,13

First differs from A355741 and A355744 at n = 35.

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.

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

The a(1469) = 5 ways are: {2,2}, {2,3}, {2,5}, {3,3}, {3,5}.

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

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

Allowing all divisors gives A355735, firsts A355736, reverse A355749.

Not requiring an increasing sequence gives A355741.

Choosing a multiset instead of weakly increasing sequence gives A355744.

A000005 counts divisors.

A001414 adds up distinct prime divisors, counted by A001221.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A120383 lists numbers divisible by all of their prime indices.

A324850 lists numbers divisible by the product of their prime indices.

A355731 chooses of a divisor of each prime index, firsts A355732.

A355733 chooses a multiset of divisors, firsts A355734.

Cf. A000720, A076610, `A335433, ~`A335448, `~A340852, A355737, A355739, `A355740, A355742.

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Gus Wiseman, Jul 18 2022

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allocated for Gus Wiseman

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