%I #26 Sep 11 2018 21:12:54

%S 1,1,1,2,1,2,1,2,2,2,1,2,2,2,1,4,2,2,2,1,4,2,2,2,2,1,2,4,2,2,2,2,2,1,

%T 2,5,4,2,2,2,2,2,1,2,5,4,2,2,2,2,2,2,1,2,5,4,2,2,2,2,2,2,1,5,2,5,4,2,

%U 2,2,2,2,2,1,5,2,5,4,2,2,4,2,2,2,2,1,5

%N Number of inequivalent leaf-colorings of the free pure symmetric multifunction 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 Inequivalent representatives of the a(441) = 11 colorings of the expression e(441) = o[o,o][o] are the following.

%e 1[1,1][1]

%e 1[1,1][2]

%e 1[1,2][1]

%e 1[1,2][2]

%e 1[1,2][3]

%e 1[2,2][1]

%e 1[2,2][2]

%e 1[2,2][3]

%e 1[2,3][1]

%e 1[2,3][2]

%e 1[2,3][4]

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 18 2018