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%I #6 Aug 20 2018 07:54:57
%S 1,4,16,128,256,16384,65536,268435456,4294967296,562949953421312,
%T 9007199254740992,72057594037927936,18446744073709551616,
%U 316912650057057350374175801344,81129638414606681695789005144064,5192296858534827628530496329220096
%N e-numbers of unlabeled rooted trees. 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 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 empty subexpressions f[] or subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.
%e The sequence contains 16384 = 2^14 = 2^(prime(1) * prime(4)) because 1 and 4 both already belong to the sequence.
%e The sequence of unlabeled rooted trees with e-numbers in the sequence begins:
%e 1: o
%e 4: (o)
%e 16: (oo)
%e 128: ((o))
%e 256: (ooo)
%e 16384: (o(o))
%e 65536: (oooo)
%e . (oo(o))
%e . (ooooo)
%e . ((o)(o))
%e ((oo))
%e (ooo(o))
%e (oooooo)
%e (o(o)(o))
%e (o(oo))
%e (oooo(o))
%e (ooooooo)
%e (oo(o)(o))
%t baQ[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,_?(And@@Cases[FactorInteger[#],{p_,k_}:>baQ[PrimePi[p]]]&)}}]];
%t Select[2^Range[0,50],baQ]
%Y A subsequence of A000079 and A318151.
%Y Cf. A000081, A007916, A052409, A052410, A277576, A277996, A280000.
%Y Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318153.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 19 2018
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