A296095 -id:A296095 - cp's OEIS frontend

A024614 Numbers of the form x^2 + xy + y^2, where x and y are positive integers.

3, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228

Offset: 1

Clark Kimberling

Equivalently, sequence A024612 with duplicates removed; i.e., numbers of the form i^2 - i*j + j^2, where 1 <= i < j.
A subsequence of A135412, which consists of multiples (by squarefree factors) of the numbers listed here. It appears that this lists numbers > 1 which have in their factorization: (a) no even power of 3 unless there is a factor == 1 (mod 6); (b) no odd power of 2 or of a prime == 5 (mod 6) and no even power unless there is a factor 3 or == 1 (mod 6). - M. F. Hasler, Aug 17 2016
If we regroup the entries in a triangle with row lengths A004526
   3,
   7,
  12, 13,
  19, 21,
  27, 28, 31,
  37, 39, 43,
... it seems that the j-th row of the triangle contains the numbers i^2+j^2-i*j in row j>=2 and column i = floor((j+1)/2) .. j-1. - R. J. Mathar, Aug 21 2016
Proof of the above characterization: the sequence is the union of 3*(the squares A000290) and A024606 (numbers x^2+xy+y^2 with x > y > 0). For the latter it is known that these are the numbers with a factor p==1 (mod 6) and any prime factor == 2 (mod 3) occurring to an even power. The former (3*n^2) are the same as (odd power of 3)*(even power of any other prime factor). The union of the two cases yields the earlier characterization. - M. F. Hasler, Mar 04 2018
Least term that can be written in exactly n ways is A300419(n). - Altug Alkan, Mar 04 2018
For the general theory see the Fouvry et al. reference and A296095. Bounds used in the Julia program are based on the theorems in this paper. - Peter Luschny, Mar 10 2018

			3 = 1^2 + 1^2 + 1*1, 7 = 2^2 + 1^2 + 2*1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Peter Luschny)
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Cf. A003136, A007645 (prime terms), A024612, A135412, A296095, A300419.

function isA024614(n)
    n % 3 >= 2 && return false
    n == 3 && return true
    M = Int(round(2*sqrt(n/3)))
    for y in 2:M, x in 1:y
        n == x^2 + y^2 + x*y && return true
    end
    return false
end
A024614list(upto) = [n for n in 1:upto if isA024614(n)]
println(A024614list(228)) # Peter Luschny, Mar 02 2018 updated Mar 17 2018

isA024614 := proc(n)
    local i,j,disc;
    # n=i^2+j^2-i*j = (j-i)^2+i*j, 1<=i=1 and i*j>=j and i^2+j^2-i*j >= 1+j max search radius
    for j from 2 to n-1 do
        # i=(j +- sqrt(4n-3j^2))/2
        disc := 4*n-3*j^2 ;
        if disc >= 0 then
            if issqr(disc) then
                i := (j+sqrt(disc))/2 ;
                if type(i,'integer') and i >= 1 and i= 1 and iA024614(t) then
        printf("%d %d\n",n,t) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 21 2016
# second Maple program:
a:= proc(n) option remember; local k, x;
      for k from a(n-1)+1 do for x while x^2 x^2+(x+y)*y=k)((isqrt(4*k-3*x^2)-x)/2) then return k fi
      od od
    end: a(0):=0:
seq(a(n), n=1..200);  # Alois P. Heinz, Mar 02 2018

max = 228; T0 = {}; xm = Ceiling[Sqrt[max]]; While[T = T0; T0 = Table[x^2 + x y + y^2, {x, 1, xm}, {y, x, xm}] // Flatten // Union // Select[#, # <= max&]&; T != T0, xm = 2 xm]; T (* Jean-François Alcover, Mar 23 2018 *)

is(n)={n>2&&!for(i=1, #n=Set(Col(factor(n)%6))/*consider prime factors mod 6*/, n[i][1]>1||next/*skip factors = 1 mod 6*/; /* odd power: ok only if p=3 */n[i][2]%2&&(n[i][1]!=3 || next) && return; /*even power: ok if there's a p==1, listed first*/ n[1][1]==1 || /*also ok if it's not a 3 and if there's a 3 elsewhere */ (n[i][1]==2 && i<#n && n[i+1][1]==3) || (n[i][1]>3 && for(j=1,i-1,n[j][1]==3 && next(2))||return))} \\ M. F. Hasler, Aug 17 2016, documented & bug fixed (following an observation by Altug Alkan) Mar 04 2018

is(n)={(n=factor(n))||return/*n=1*/; /*exponents % 2, primes % 3*/ n[,2]%=2; n[,1]%=3; (n=Set(Col(n))) /*odd power of a prime == 2? will be last*/ [#n][2] && n[#n][1]==2 && return; /*factor == 1? will be 1st or after 3*/ n[1+(!n[1][1] && #n>1)][1]==1 || /*thrice a square?*/ (!n[1][1]&&n[1][2]&&!for(i=2,#n,n[i][2]&&return))} \\ Alternate code, 5% slower, maybe a bit less obscure. - M. F. Hasler, Mar 04 2018

N=228; v=vector(N);
for(x=1,N, x2=x*x; if(x2>N,break); for(y=x,N, t=x2+y*(x+y); if(t>N,break); v[t]=1;));
for(n=1,N,if(v[n],print1(n,", "))); \\ Joerg Arndt, Mar 10 2018

list(lim)=my(v=List(), x2); lim\=1; for(x=1, sqrtint(4*lim\3), x2=x^2; for(y=1, min((sqrt(4*lim-3*x2)-x)/2, x), listput(v, y*(x+y)+x2))); Set(v) \\ Charles R Greathouse IV, Mar 23 2018

Edited by M. F. Hasler, Aug 17 2016

b-file for values a(1)..a(10^4) double-checked with PARI code by M. F. Hasler, Mar 04 2018

A293654 Integers not represented by cyclotomic binary forms.

Original entry on oeis.org

1, 2, 6, 14, 15, 22, 23, 24, 30, 33, 35, 38, 42, 44, 46, 47, 51, 54, 56, 59, 60, 62, 66, 69, 70, 71, 77, 78, 83, 86, 87, 88, 92, 94, 95, 96, 99, 102, 105, 107, 110, 114, 115, 118, 119, 120, 123, 126, 131, 132, 134, 135, 138, 140, 141, 142, 143, 150

Offset: 1

Michel Waldschmidt, Feb 16 2018

nonn

Possibly a supersequence of A055039. - C. S. Davis, May 10 2025

Michel Waldschmidt, Table of n, a(n) for n = 1..249
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Complement of A296095.

For n>2, subsequence of A383785, following from Proposition 6.2 of Fouvry et al.

g := 1;
for m from 1 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 (F[n] = m, max(abs(x), abs(y)) > 1
                then r[g] := m; m := m+1; n := 3;
                     x := -50; y := -50; g := g+1
                fi;
od; od; od; od;
for t to 519 do print(r[{t}] = r[t]) od;
s[1] := 1; s[2] := 2; g := 2;
for i from 1 to 518 do
    for j from r[i]+1 to r[i+1]-1 do
        g := g+1; s[g] := j
od; od;
for t to 481 do s[t] od;

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[150], !isA296095[#]&] (* Jean-François Alcover, Jun 21 2018, after Peter Luschny *)

def A293654list(upto):
    return [n for n in (1..upto) if not isA296095(n)]
print(A293654list(150)) # Peter Luschny, Feb 25 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]

cp's OEIS Frontend

A024614 Numbers of the form x^2 + xy + y^2, where x and y are positive integers.

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A293654 Integers not 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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