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%I A318151 #10 Sep 26 2019 08:10:55
%S A318151 1,2,4,8,16,64,128,256,512,4096,16384,65536,262144,524288,2097152,
%T A318151 16777216,134217728,268435456,4294967296,68719476736,274877906944,
%U A318151 4398046511104,281474976710656,562949953421312,9007199254740992,18014398509481984,72057594037927936
%N A318151 e-numbers of unlabeled rooted trees with empty leaves o[] allowed. A number n is in the sequence iff n = 2^(prime(y_1) * ... * prime(y_k)) for some k >= 0 and y_1, ..., y_k already in the sequence.
%C A318151 If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) 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). The sequence consists of all numbers n such that e(n) contains no subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.
%Y A318151 A subsequence of A000079.
%Y A318151 Cf. A000081, A007916, A029856, A052409, A052410, A277576, A277996, A280000.
%Y A318151 Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152, A318153.
%K A318151 nonn
%O A318151 1,2
%A A318151 _Gus Wiseman_, Aug 19 2018

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