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%I #6 Jul 19 2022 08:04:07
%S 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,
%T 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,
%U 1,0,2,0,2,0,1,0,1,0,2,0,1,0,1,0,1,0,2
%N Number 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.
%C First differs from A355741 and A355744 at n = 35.
%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 prime indices of 1469 are {6,30}, and there are five valid choices: (2,2), (2,3), (2,5), (3,3), (3,5), so a(1469) = 5.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Length[Select[Tuples[Union/@primeMS/@primeMS[n]],LessEqual@@#&]],{n,100}]
%Y Allowing all divisors gives A355735, firsts A355736, reverse A355749.
%Y Not requiring an increasing sequence gives A355741.
%Y Choosing a multiset instead of sequence gives A355744.
%Y A000005 counts divisors.
%Y A001414 adds up distinct prime divisors, counted by A001221.
%Y A003963 multiplies together the prime indices of n.
%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A120383 lists numbers divisible by all of their prime indices.
%Y A324850 lists numbers divisible by the product of their prime indices.
%Y A355731 chooses of a divisor of each prime index, firsts A355732.
%Y A355733 chooses a multiset of divisors, firsts A355734.
%Y Cf. A000720, A076610, A335433, A340852, A355737, A355739, A355740, A355742.
%K nonn
%O 1,13
%A _Gus Wiseman_, Jul 18 2022
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