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A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

%I A355733 #16 Apr 01 2025 10:27:03
%S A355733 1,1,2,1,2,2,3,1,3,2,2,2,4,3,4,1,2,3,4,2,5,2,3,2,3,4,4,3,4,4,2,1,4,2,
%T A355733 6,3,6,4,7,2,2,5,4,2,6,3,4,2,6,3,4,4,5,4,4,3,7,4,2,4,6,2,7,1,7,4,2,2,
%U A355733 6,6,6,3,4,6,6,4,6,7,4,2,5,2,2,5,4,4,7
%N A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.
%C A355733 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 A355733 Robert Price, <a href="/A355733/b355733.txt">Table of n, a(n) for n = 1..2000</a>
%H A355733 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A355733 The a(15) = 4 multisets are: {1,1}, {1,2}, {1,3}, {2,3}.
%e A355733 The a(18) = 3 multisets are: {1,1,1}, {1,1,2}, {1,2,2}.
%t A355733 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355733 Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,100}]
%Y A355733 Counting all choices of divisors gives A355731, firsts A355732.
%Y A355733 Positions of first appearances are A355734.
%Y A355733 Choosing weakly increasing divisors gives A355735, firsts A355736.
%Y A355733 Choosing only prime divisors gives A355744.
%Y A355733 The version choosing a divisor of each number from 1 to n is A355747.
%Y A355733 A000005 counts divisors.
%Y A355733 A001414 adds up distinct prime divisors, counted by A001221.
%Y A355733 A003963 multiplies together the prime indices of n.
%Y A355733 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A355733 A061395 selects the maximum prime index.
%Y A355733 A120383 lists numbers divisible by all of their prime indices.
%Y A355733 A324850 lists numbers divisible by the product of their prime indices.
%Y A355733 A340852 lists numbers that can be factored into divisors of bigomega.
%Y A355733 Cf. A000720, A076610, A289509, A316524, A335433, A335448, A340606, A344616, A355737, A355739, A355740, A355741, A355746.
%K A355733 nonn
%O A355733 1,3
%A A355733 _Gus Wiseman_, Jul 16 2022