A318150 - OEIS

%I #13 Feb 24 2019 14:29:32

%S 1,4,36,128,2025,21025,279936,4338889,449482401,78701569444,

%T 373669453125,18845583322500,1347646586640625,202054211912421649,

%U 6193981883008128893161,139629322539586311507076,170147232533595290155627,355156175404848064835984400

%N e-numbers of free pure functions with one atom.

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

%H Charlie Neder, <a href="/A318150/b318150.txt">Table of n, a(n) for n = 1..44</a>

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

%e The sequence of all free pure functions with one atom together with their e-numbers begins:

%e         1: o

%e         4: o[o]

%e        36: o[o][o]

%e       128: o[o[o]]

%e      2025: o[o][o][o]

%e     21025: o[o[o]][o]

%e    279936: o[o][o[o]]

%e   4338889: o[o][o][o][o]

%Y A subsequence of A001597.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 19 2018

%E More terms from _Charlie Neder_, Feb 23 2019