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%I #18 Sep 11 2018 21:16:23
%S 1,2,3,2,4,3,5,3,3,4,6,4,4,5,7,2,5,5,6,8,3,6,6,7,4,9,3,4,7,7,8,4,5,10,
%T 4,3,5,8,8,9,5,6,11,5,4,6,9,9,5,10,6,7,12,6,5,7,10,10,6,11,7,8,13,3,7,
%U 6,8,11,11,7,12,8,9,14,4,8,7,9,12,12,3,8
%N Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
%C If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).
%e The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.
%t nn=1000;
%t radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
%t rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
%t Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
%t exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
%t Table[Length[Union[Cases[exp[n],_,{0,Infinity},Heads->True]]],{n,100}]
%Y Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 18 2018
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