A293654 -id:A293654 - cp's OEIS frontend

A296095 Integers represented by cyclotomic binary forms.

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

Offset: 1

Michel Waldschmidt, Feb 14 2018

nonn

Possibly a subsequence of A000401. - C. S. Davis, May 10 2025

All terms divisible by 11 appear to be either of the form 11^2*A383784(n) for n>1 or x^4 + u*x^3*y + x^2*y^2 + u*x*y^3 + y^4 for x>y>0 and u={-1, 1}. - C. S. Davis, May 14 2025

Peter Luschny, Table of n, a(n) for n = 1..1000 (terms 1..519 from Michel Waldschmidt).
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Complement of A293654.

Supersequence of A383784(n) for n>3, according to Proposition 6.2 of Fouvry et al.

using Nemo
function isA296095(n)
    n < 3 && return false
    R, z = PolynomialRing(ZZ, "z")
    N = QQ(n)
    # Bounds from Fouvry, Levesque and 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

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;

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-François Alcover, Feb 20 2018, translated from Peter Luschny's Sage script, updated Mar 01 2018 *)

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)
            for x in (1..y):
                if n == p(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

A299214 Number of representations of integers by cyclotomic binary forms.

Original entry on oeis.org

0, 0, 8, 16, 8, 0, 24, 4, 16, 8, 8, 12, 40, 0, 0, 40, 16, 4, 24, 8, 24, 0, 0, 0, 24, 8, 12, 24, 8, 0, 32, 8, 0, 8, 0, 16, 32, 0, 24, 8, 8, 0, 32, 0, 8, 0, 0, 12, 40, 12, 0, 32, 8, 0, 8, 0, 32, 8, 0, 0, 48, 0, 24, 40, 16, 0, 24, 8, 0, 0, 0, 4, 48, 8, 12, 24

Offset: 1

Michel Waldschmidt, Feb 16 2018

nonn

a(m) is the number of solutions of the equation Phi_n(x,y) = m with n >= 3 and max{|x|,|y|} >= 2. Here the binary form Phi_n(x,y) is the homogeneous version of the cyclotomic polynomial phi_n(t).

One can prove that a(m) is always a multiple of 4.

Michel Waldschmidt, Table of n, a(n) for n = 1..1000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

The sequence of indices m with a(m) != 0 is A296095.

The sequence of indices m with a(m) = 0 is A293654.

using Nemo
function countA296095(n)
    if n < 3 return 0 end
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))        # Fouvry & Levesque & Waldschmidt
    N = QQ(n); count = 0
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 1:M, j in 0:M if max(j, m) > 1
            N == m^e*subst(c, QQ(j,m)) && (count += 1)
    end end end
    4*count
end
A299214list(upto) = [countA296095(n) for n in 1:upto]
print(A299214list(76)) # Peter Luschny, Feb 25 2018

x := 'x'; y := 'y':
with(numtheory): for n from 3 to 1000 do
F[n] := expand(y^phi(n)*cyclotomic(n, x/y))  od:
g := 0:
for m from 1 to 1000 do
   for n from 3 to 60 do  # For the bounds see the reference.
      for x from -60 to 60 do
         for y from -60 to 60 do
            if F[n] = m and  max(abs(x), abs(y)) > 1
                then g := g+1 fi:
         od:
      od:
   od: a[m] := g: print(m, a[m]): g := 0
od:

For[n = 3, n <= 100, n++, F[n] = Expand[y^EulerPhi[n] Cyclotomic[n, x/y]]]; g = 0; For[m = 1, m <= 100, m++, For[n = 3, n <= 60, n++, For[x = -60, x <= 60, x++, For[y = -60, y <= 60, y++, If[F[n] == m && Max[Abs[x], Abs[y] ] > 1, g = g+1]]]]; a[m] = g; Print[m, " ", a[m]]; g = 0];
Array[a, 100] (* Jean-François Alcover, Dec 01 2018, from Maple *)

A299733 Prime numbers represented in more than one way by cyclotomic binary forms f(x,y) with x and y prime numbers and y < x.

Original entry on oeis.org

19, 97, 33751

Offset: 1

Peter Luschny, Feb 21 2018

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.

There are only three prime numbers below 600000 which satisfy the given conditions. No prime number below 600000 exists which has more than one representation if we require a representation by odd prime numbers y < x.

			33751 = f(131,79) for f(x,y) = x^2 + x*y + y^2.
33751 = f( 13, 2) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.

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

Subsequence of A299929.

Cf. A293654, A296095, A299214, A299498, A299928, A299930, A299956, A299964.

using Nemo
function isA299733(n)
    if n < 3 || !isprime(ZZ(n)) return false end
    R, x = PolynomialRing(ZZ, "x")
    K = floor(Int, 5.383*log(n)^1.161) # Bounds from
    M = floor(Int, 2*sqrt(n/3)) # Fouvry & Levesque & Waldschmidt
    N = QQ(n); multi = 0
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 2:M if isprime(ZZ(m))
            for j in m:M if isprime(ZZ(j))
                if N == m^e*subst(c, QQ(j,m)) multi += 1
    end end end end end end
    multi > 1
end # Peter Luschny, May 16 2019

A299733(upto) =
{
    my(K, M, phi, multi);
    forprime(n = 2, upto, multi = 0;
        K = floor(5.383*log(n)^1.161);
        M = floor(2*sqrt(n/3));
        for(k = 3, K,
            phi = eulerphi(k);
            forprime(y = 2, M,
                forprime(x = y + 1, M,
                    if(n == y^phi*polcyclo(k, x/y),
                        multi += 1
                    )
                )
            )
        );
        if(multi > 1, print(n," has multiple reps!"))
    )
}
A299733(100000)

A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Feb 25 2018

nonn

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.

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

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

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))

A299498 Integers primitively represented by cyclotomic binary forms.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 17, 19, 21, 25, 26, 29, 31, 34, 37, 39, 41, 43, 49, 50, 53, 55, 57, 58, 61, 65, 67, 73, 74, 79, 82, 85, 89, 91, 93, 97, 101, 103, 106, 109, 111, 113, 121, 122, 125, 127, 129, 130, 133, 137, 139, 145, 146, 147, 149, 151, 157, 163, 169, 170

Offset: 1

Peter Luschny, Feb 25 2018

nonn

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 primitively represented by f if f(x,y) = n has an integer solution such that x is prime to y.

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

Cf. A293654, A296095, A299214, A299733, A299928, A299929, A299930, A299956, A299964.

using Nemo
function isA299498(n)
    isPrimeTo(n, k) = gcd(ZZ(n), ZZ(k)) == ZZ(1)
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
    N = QQ(n)
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 1:M, j in m+1:M if isPrimeTo(m, j)
            N == m^e*subst(c, QQ(j,m)) && return true
    end end end
    return false
end
A299498list(upto) = [n for n in 1:upto if isA299498(n)]
print(A299498list(170))

A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.

Original entry on oeis.org

7, 13, 19, 29, 34, 37, 39, 49, 53, 55, 58, 61, 67, 74, 79, 91, 93, 97, 103, 109, 125, 127, 129, 130, 139, 146, 147, 163, 170, 173, 178, 194, 199, 201, 211, 217, 218, 219, 223, 229, 237, 247, 259, 273, 277, 283, 290, 291, 293, 298, 309, 313, 314, 327, 338, 349

Offset: 1

Peter Luschny, Feb 21 2018

nonn

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.

			There are exactly four ways to represent 13 by a cyclotomic binary form f(x,y) if we require x > y > 0. In one case, x and y are prime.
13 = f(2, 1) where f(x, y) = x^4 - x^2*y^2 + y^4,
13 = f(3, 1) where f(x, y) = x^2 + x*y + y^2,
13 = f(3, 2) where f(x, y) = x^2 + y^2,
13 = f(4, 3) where f(x, y) = x^2 - x*y + y^2.

Trygve Nagell, Sur les représentations de l’unité par les formes binaires biquadratiques du premier rang, Arkiv för Mat. 5 (6), (1965), 477-521, (p. 517).

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

Cf. A299929 (represented primes), A293654, A296095, A299214, A299498, A299733, A299930, A299956, A299964.

using Nemo
function isA299928(n)
    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(2), x in P(y+1)
            N == y^e*subst(c, QQ(x, y)) && return true
        end
    end
    return false
end
A299928list(upto) = [n for n in 1:upto if isA299928(n)]
println(A299928list(350))

A299956 Prime numbers not in A299930.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 101, 103, 107, 113, 131, 137, 149, 151, 157, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 331, 337, 347, 353, 359

Offset: 1

Peter Luschny, Feb 25 2018

nonn

These prime numbers cannot be represented by a cyclotomic binary form f as p = f(x,y) with x and y odd prime numbers and x > y. Apart from 2 they appear as the leafs of cbf-trees introduced in A299930.

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299930, A299964.

isA299956(n) = isprime(ZZ(n)) && !isA299930(n)
A299956list(upto) = [n for n in 1:upto if isA299956(n)]
println(A299956list(1000))

A299964 Integers represented in more than one way by a cyclotomic binary form f(x,y) where x and y are prime numbers and 0 < y < x.

Original entry on oeis.org

19, 39, 97, 147, 247, 259, 327, 399, 410, 427, 481, 650, 777, 890, 903, 1010, 1027, 1130, 1209, 1267, 1443, 1490, 1533, 1677, 1730, 1767, 1802, 1813, 1898, 1911, 1970, 2037, 2119, 2210, 2330, 2378, 2667, 2793, 2847, 3050, 3170, 3297, 3367, 3477, 3530, 3603

Offset: 1

Peter Luschny, Feb 25 2018

nonn

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 in this sequence if f(x,y) = n has more than one integer solution where f is a cyclotomic binary form and x and y are prime numbers with 0 < y < x.

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

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299930, A299956.

function countA299928(n)
    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); count = 0
    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(2), x in P(y+1)
            if N == y^e*subst(c, QQ(x, y))
                count += 1
    end end end
    return count
end
A299964list(upto) = [n for n in 1:upto if countA299928(n) > 1]
println(A299964list(3640))

A299929 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y prime numbers and 0 < y < x.

Original entry on oeis.org

7, 13, 19, 29, 37, 53, 61, 67, 79, 97, 103, 109, 127, 139, 163, 173, 199, 211, 223, 229, 277, 283, 293, 313, 349, 397, 421, 433, 439, 457, 463, 487, 541, 577, 607, 641, 643, 691, 727, 733, 739, 787, 877, 937, 997, 1009, 1031, 1063, 1093, 1327, 1373, 1423, 1447

Offset: 1

Peter Luschny, Feb 21 2018

nonn

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.

			6841 = f(7,5) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.

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

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299930, A299956, A299964.

A299929list(upto) = [n for n in 1:upto if isprime(ZZ(n)) && isA299928(n)]
println(A299929list(1450))

isA299929[n_] := If[! PrimeQ[n], Return[False],
   K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
   For[k = 3, k <= K, k++,
   For[y = 1, y <= M, y++, If[PrimeQ[y], For[x = y + 1, x <= M, x++, If[PrimeQ[x],
   If[n == y^EulerPhi[k] Cyclotomic[k, x/y], Return[True]]]]]]];
Return[False]]; Select[Range[1450], isA299929]

A325143 Primes represented by cyclotomic binary forms.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 109, 113, 127, 137, 139, 149, 151, 157, 163, 173, 181, 193, 197, 199, 211, 223, 229, 233, 241, 257, 269, 271, 277, 281, 283, 293, 307, 313, 317, 331, 337, 349, 353, 367, 373

Offset: 1

Peter Luschny, May 16 2019

nonn

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.

Peter Luschny, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, Acta Arithmetica 184 (2018), 67-86; arXiv:1712.09019, arXiv:1712.09019 [math.NT], 2017.

Subsequence of A296095. Complement A325145. Number of A325141.

Cf. A293654, A299214, A299498, A299733, A299928, A299930, A299956, A299964.

using Nemo
function isA325143(n)
    (n < 3 || !isprime(ZZ(n))) && return false
    R, x = PolynomialRing(ZZ, "x")
    K = floor(Int, 5.383*log(n)^1.161) # Bounds from
    M = floor(Int, 2*sqrt(n/3)) # Fouvry & Levesque & Waldschmidt
    N = QQ(n)
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 1:M, j in 0:M if max(j, m) > 1
            N == m^e*subst(c, QQ(j,m)) && return true
    end end end
    return false
end
[n for n in 1:373 if isA325143(n)] |> println

cp's OEIS Frontend

A296095 Integers represented by cyclotomic binary forms.

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A299214 Number of representations of integers by cyclotomic binary forms.

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A299733 Prime numbers represented in more than one way by cyclotomic binary forms f(x,y) with x and y prime numbers and y < x.

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A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.

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A299498 Integers primitively represented by cyclotomic binary forms.

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A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.

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A299956 Prime numbers not in A299930.

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A299964 Integers represented in more than one way by a cyclotomic binary form f(x,y) where x and y are prime numbers and 0 < y < x.

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