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%I #8 Jul 19 2022 08:04:16
%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 1,0,2,0,2,0,1,0,2,0,1,0,2,0,1,0,1,0,1,0,1,0,1,0,1,0,2,0,1,0,2,0,1,0,
%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 sequence of prime factors, one of each prime index of n.
%C First differs from A355744 at a(169) = 4, A355744(169) = 3.
%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>.
%F Totally multiplicative with a(prime(k)) = A001221(k).
%e The prime indices of 1131 are {2,6,10}, and the a(1131) = 4 choices are: {2,2,2}, {2,2,5}, {2,3,2}, {2,3,5}.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Times@@PrimeNu/@primeMS[n],{n,100}]
%Y Positions of 0's are A299174.
%Y The version for all divisors is A355731, firsts A355732.
%Y Choosing prime-power divisors gives A355742.
%Y Positions of 1's are A355743.
%Y Counting multisets instead of sequences gives A355744.
%Y The weakly increasing case is A355745, all divisors A355735.
%Y A001414 adds up distinct prime factors, 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 A289509 lists numbers with relatively prime prime indices.
%Y A324850 lists numbers divisible by the product of their prime indices.
%Y Cf. A000720, A061395, A076610, A120383, A335433, A355733, A355737, A355739, A355746, A355747.
%K nonn,mult
%O 1,13
%A _Gus Wiseman_, Jul 18 2022
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