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%I A317056 #43 Sep 11 2018 21:16:02
%S A317056 0,1,2,1,3,2,4,2,2,3,5,3,3,4,6,1,4,4,5,7,2,5,5,6,3,8,2,3,6,6,7,3,4,9,
%T A317056 3,2,4,7,7,8,4,5,10,4,3,5,8,8,4,9,5,6,11,5,4,6,9,9,5,10,6,7,12,2,6,5,
%U A317056 7,10,10,6,11,7,8,13,3,7,6,8,11,11,2,7,12
%N A317056 Depth of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
%C A317056 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 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 A317056 e(21025) = o[o[o]][o] has depth 3 so a(21025) = 3.
%t A317056 nn=1000;
%t A317056 radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
%t A317056 rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
%t A317056 Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
%t A317056 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 A317056 Table[Max@@Length/@Position[exp[n],_],{n,200}]
%Y A317056 Cf. A007916, A052409, A052410, A109082, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.
%K A317056 nonn
%O A317056 1,3
%A A317056 _Gus Wiseman_, Aug 18 2018

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