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A318150
e-numbers of free pure functions with one atom.
5
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
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]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 19 2018
EXTENSIONS
More terms from Charlie Neder, Feb 23 2019
STATUS
approved