login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y. 9
19, 37, 79, 97, 109, 127, 139, 163, 223, 229, 277, 283, 313, 349, 397, 421, 433, 439, 457, 607, 643, 691, 727, 733, 739, 877, 937, 997, 1063, 1093, 1327, 1423, 1459, 1489, 1567, 1579, 1597, 1627, 1657, 1699, 1753, 1777, 1801, 1987, 1999, 2017, 2089, 2113, 2203 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A cyclotomic binary form over Z is a homogeneous polynomial in two variables which has the form f(x, y) = y^EulerPhi(k)*CyclotomicPolynomial(k, x/y) where k is some integer >= 3. An integer n is represented by f if f(x,y) = n has an integer solution.

We say a prime number p decomposes into x and y if x and y are odd prime numbers and there exists a cyclotomic binary form f such that p = f(x,y). The transitive closure of this relation can be displayed as a binary tree, the cbf-tree of p. A cbf-tree is squarefree if all its leafs are distinct. Examples are:

.

      33751                   23833                   310567

       / \                     /  \                    /  \

    131   79                163    19               359    283

          / \               / \    / \                     / \

         7   3            11   3  5   3                  19   13

                                                        /  \

                                                       5    3

.

The leafs of these trees are in A299956. Related to the question whether the root of a cbf-tree can be reconstructed from its leafs is A299733.

LINKS

Table of n, a(n) for n=1..49.

Etienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

PROG

(Julia)

using Nemo

function isA299930(n)

    !isprime(ZZ(n)) && return false

    R, z = PolynomialRing(ZZ, "z")

    K = Int(floor(5.383*log(n)^1.161)) # Bounds from

    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt

    N = QQ(n)

    P(u) = (p for p in u:M if isprime(ZZ(p)))

    for k in 3:K

        e = Int(eulerphi(ZZ(k)))

        c = cyclotomic(k, z)

        for y in P(3), x in P(y+2)

            N == y^e*subst(c, QQ(x, y)) && return true

    end end

    return false

end

A299930list(upto) = [n for n in 1:upto if isA299930(n)]

println(A299930list(2203))

CROSSREFS

Cf. A299956 (complement), A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299956, A299964.

Sequence in context: A245381 A050528 A257074 * A053685 A244111 A136063

Adjacent sequences:  A299927 A299928 A299929 * A299931 A299932 A299933

KEYWORD

nonn

AUTHOR

Peter Luschny, Feb 25 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 28 12:47 EDT 2022. Contains 357070 sequences. (Running on oeis4.)