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%I A355731 #17 Jul 24 2022 14:13:46
%S A355731 1,1,2,1,2,2,3,1,4,2,2,2,4,3,4,1,2,4,4,2,6,2,3,2,4,4,8,3,4,4,2,1,4,2,
%T A355731 6,4,6,4,8,2,2,6,4,2,8,3,4,2,9,4,4,4,5,8,4,3,8,4,2,4,6,2,12,1,8,4,2,2,
%U A355731 6,6,6,4,4,6,8,4,6,8,4,2,16,2,2,6,4,4
%N A355731 Number of ways to choose a sequence of divisors, one of each element of the multiset of prime indices of n (row n of A112798).
%C A355731 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 A355731 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cartesian_product">Cartesian product</a>.
%F A355731 a(n) = Product_{k=1..A001222(n)} A000005(A112798(n,k)).
%e A355731 The a(15) = 4 choices are: (1,1), (1,3), (2,1), (2,3).
%e A355731 The a(18) = 4 choices are: (1,1,1), (1,1,2), (1,2,1), (1,2,2).
%t A355731 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355731 Table[Times@@Length/@Divisors/@primeMS[n],{n,100}]
%Y A355731 Positions of 1's are A000079.
%Y A355731 Dominated by A003963 (cf. A049820), with equality at A003586.
%Y A355731 Positions of first appearances are A355732.
%Y A355731 Counting distinct sequences after sorting gives A355733, firsts A355734.
%Y A355731 Requiring the result to be weakly increasing gives A355735, firsts A355736.
%Y A355731 Requiring the result to be relatively prime gives A355737, firsts A355738.
%Y A355731 Requiring the choices to be distinct gives A355739, zeros A355740.
%Y A355731 For prime divisors A355741, prime-powers A355742, weakly increasing A355745.
%Y A355731 Choosing divisors of each of 1..n and resorting gives A355747.
%Y A355731 An ordered version (using standard order compositions) is A355748.
%Y A355731 A000005 counts divisors.
%Y A355731 A001414 adds up distinct prime divisors, counted by A001221.
%Y A355731 A003963 multiplies together the prime indices of n.
%Y A355731 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A355731 A120383 lists numbers divisible by all of their prime indices.
%Y A355731 A289509 lists numbers with relatively prime prime indices.
%Y A355731 A324850 lists numbers divisible by the product of their prime indices.
%Y A355731 A340852 lists numbers that can be factored into divisors of bigomega.
%Y A355731 Cf. A000720, A061395, A076610, A316524, A326841, A335433, A335448, A340606, A344616, A345957, A355744, A355746.
%K A355731 nonn
%O A355731 1,3
%A A355731 _Gus Wiseman_, Jul 16 2022

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