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.

1, 2, 4, 8, 16, 64, 128, 256, 512, 4096, 16384, 65536, 262144, 524288, 2097152, 16777216, 134217728, 268435456, 4294967296, 68719476736, 274877906944, 4398046511104, 281474976710656, 562949953421312, 9007199254740992, 18014398509481984, 72057594037927936

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..27.

Crossrefs

A subsequence of A000079.

Cf. A000081, A007916, A029856, A052409, A052410, A277576, A277996, A280000.

Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152, A318153.

Sequence in context: A041013 A018754 A264980 * A076086 A246907 A074700

Adjacent sequences: A318148 A318149 A318150 * A318152 A318153 A318154

Keyword

nonn

Author

Gus Wiseman, Aug 19 2018

Status

approved