A318152 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.

1, 4, 16, 128, 256, 16384, 65536, 268435456, 4294967296, 562949953421312, 9007199254740992, 72057594037927936, 18446744073709551616, 316912650057057350374175801344, 81129638414606681695789005144064, 5192296858534827628530496329220096

Offset: 1

Author

Gus Wiseman, Aug 19 2018

Keywords

• nonn

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 empty subexpressions f[] or subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.

Examples

			The sequence contains 16384 = 2^14 = 2^(prime(1) * prime(4)) because 1 and 4 both already belong to the sequence.
The sequence of unlabeled rooted trees with e-numbers in the sequence begins:
      1: o
      4: (o)
     16: (oo)
    128: ((o))
    256: (ooo)
  16384: (o(o))
  65536: (oooo)
    .    (oo(o))
    .    (ooooo)
    .    ((o)(o))
         ((oo))
         (ooo(o))
         (oooooo)
         (o(o)(o))
         (o(oo))
         (oooo(o))
         (ooooooo)
         (oo(o)(o))
		

Crossrefs

A subsequence of A000079 and A318151.

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

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

Programs

Mathematica

baQ[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,?(And@@Cases[FactorInteger[#],{p,k_}:>baQ[PrimePi[p]]]&)}}]];
Select[2^Range[0,50],baQ]