login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2017 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296095 Integers represented by cyclotomic binary forms. 12
3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 41, 43, 45, 48, 49, 50, 52, 53, 55, 57, 58, 61, 63, 64, 65, 67, 68, 72, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 89, 90, 91, 93, 97, 98, 100, 101, 103, 104, 106, 108, 109, 111, 112, 113, 116, 117, 121, 122 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Michel Waldschmidt and Peter Luschny (Michel Waldschmidt to 519), Table of n, a(n) for n = 1..1000

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

MAPLE

with(numtheory): for n from 3 to 1000 do F[n] := expand(y^phi(n)*cyclotomic(n, x/y)) od: for m to 1000 do for n from 3 to 50 do for x from -50 to 50 do for y from -50 to 50 do if `and`(F[n] = m, max(abs(x), abs(y)) > 1) then print(m); m := m+1; n := 3; x := -50; y := -50 end if end do end do end do end do;

MATHEMATICA

isA296095[n_]:=

If[n<3, Return[False],

logn = Log[n]^1.161;

K = Floor[5.383*logn];

M = Floor[2*(n/3)^(1/2)];

k = 3;

While[True,

   If[k==7,

      K = Ceiling[4.864*logn];

      M = Ceiling[2*(n/11)^(1/4)]

   ];

   For[y=2, y<=M, y++,

      p[z_] = y^EulerPhi[k]*Cyclotomic[k, z];

      For[x=1, x<=y, x++, If[n==p[x/y], Return[True]]]

   ];

   k++;

   If[k>K, Break[]]

];

Return[False]

];

Select[Range[122], isA296095] (* Jean-Fran├žois Alcover, Feb 20 2018, translated from Peter Luschny's Sage script, updated Mar 01 2018 *)

PROG

(Sage)

var('z')

def isA296095(n):

    if n < 3: return false

    logn = log(n)^1.161

    K = floor(5.383*logn)

    M = floor(2*(n/3)^(1/2))

    k = 3

    while true:

        if k == 7:

            K = ceil(4.864*logn)

            M = ceil(2*(n/11)^(1/4))

        for y in (2..M):

            p = y^euler_phi(k)*cyclotomic_polynomial(k, z)

            for x in (1..y):

                if n == p.substitute(z = x/y): return true

        k += 1

        if k > K: break

    return false

def A296095list(upto):

    return [n for n in (1..upto) if isA296095(n)]

print(A296095list(122)) # Peter Luschny, Feb 28 2018

(Julia)

using Nemo

function isA296095(n)

    n < 3 && return false

    R, z = PolynomialRing(ZZ, "z")

    N = QQ(n)

    # Bounds from Fouvry & Levesque & Waldschmidt

    logn = log(n)^1.161

    K = Int(floor(5.383*logn))

    M = Int(floor(2*(n/3)^(1/2)))

    k = 3

    while true

        c = cyclotomic(k, z)

        e = Int(eulerphi(ZZ(k)))

        if k == 7

            K = Int(ceil(4.864*logn))

            M = Int(ceil(2*(n/11)^(1/4)))

        end

        for y in 2:M, x in 1:y

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

        end

        k += 1

        k > K && break

    end

    return false

end

A296095list(upto) = [n for n in 1:upto if isA296095(n)]

println(A296095list(2040)) # Peter Luschny, Feb 28 2018

CROSSREFS

Complement of A293654.

Sequence in context: A063992 A171520 A039233 * A075748 A039177 A058986

Adjacent sequences:  A296092 A296093 A296094 * A296096 A296097 A296098

KEYWORD

nonn

AUTHOR

Michel Waldschmidt, Feb 14 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 20:24 EST 2018. Contains 318023 sequences. (Running on oeis4.)