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

 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 leaves 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. Étienne 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: A050528 A257074 A357935 * 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.

Last modified August 12 12:24 EDT 2024. Contains 375092 sequences. (Running on oeis4.)