A317056 - OEIS

%I #43 Sep 11 2018 21:16:02

%S 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 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 7,10,10,6,11,7,8,13,3,7,6,8,11,11,2,7,12

%N Depth 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 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 e(21025) = o[o[o]][o] has depth 3 so a(21025) = 3.

%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[Max@@Length/@Position[exp[n],_],{n,200}]

%Y Cf. A007916, A052409, A052410, A109082, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 18 2018