A318150 e-numbers of free pure functions with one atom.

1, 4, 36, 128, 2025, 21025, 279936, 4338889, 449482401, 78701569444, 373669453125, 18845583322500, 1347646586640625, 202054211912421649, 6193981883008128893161, 139629322539586311507076, 170147232533595290155627, 355156175404848064835984400

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). This sequence consists of all numbers n such that e(n) contains no non-unitary subexpressions f[x_1, ..., x_k] where k != 1.

Links

Charlie Neder, Table of n, a(n) for n = 1..44

Formula

a(1) = 1, and if a and b are in this sequence then so is rad(a)^prime(b). - Charlie Neder, Feb 23 2019

Example

The sequence of all free pure functions with one atom together with their e-numbers begins:
        1: o
        4: o[o]
       36: o[o][o]
      128: o[o[o]]
     2025: o[o][o][o]
    21025: o[o[o]][o]
   279936: o[o][o[o]]
  4338889: o[o][o][o][o]

Crossrefs

A subsequence of A001597.

Cf. A000108, A007916, A052409, A052410, A277576, A277996, A280000.

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

Sequence in context: A144298 A072109 A045490 * A275133 A374387 A060783

Adjacent sequences: A318147 A318148 A318149 * A318151 A318152 A318153

Keyword

nonn

Author

Gus Wiseman, Aug 19 2018

Extensions

More terms from Charlie Neder, Feb 23 2019

Status

approved