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%I #16 Jul 15 2023 10:36:18
%S 1,2,1,3,4,2,5,1,6,7,8,3,9,1,4,10,11,2,12,13,14,5,15,16,6,3,17,1,18,7,
%T 2,19,20,21,22,8,23,1,24,9,4,25,26,27,10,28,29,30,5,11,31,3,32,12,33,
%U 34,1,35,36,13,6,37,2,14,38,39,15,40,41,1,42,7,4,43
%N Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%F a(n) = A358104(n)/A358105(n).
%e The 12th divisible pair is (2,6) so a(12) = 3.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y/x,{0}],{n,100}]
%Y The divisible pairs are ranked by A318990, proper A339005.
%Y Quotient of A358104 and A358105.
%Y A different ordering is A358106.
%Y A000040 lists the primes.
%Y A001222 counts prime indices, distinct A001221.
%Y A001358 lists the semiprimes, squarefree A006881.
%Y A003963 multiplies together prime indices.
%Y A056239 adds up prime indices.
%Y A358192/A358193 gives quotients of semiprime indices.
%Y Cf. A000720, A027751, A032741, A215366, A289508, A289509, A296150, A300912, A318991, A318992.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 02 2022
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