A034017 -id:A034017 - cp's OEIS frontend

A250310 Numbers whose squares are of the form x^2 + y^2 + 3 where x >= y >= 0 (repetitions omitted).

2, 4, 8, 10, 14, 20, 22, 26, 32, 34, 40, 44, 46, 52, 56, 58, 64, 68, 74, 80, 86, 88, 92, 94, 98, 100, 110, 112, 118, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 158, 164, 172, 178, 184, 190, 194, 202, 206, 208, 212, 218, 220, 230, 238, 242, 244, 250, 254, 256, 266, 268, 274, 278, 290, 296, 298

Offset: 1

Frank M Jackson and Stalislav Takhaev, Jan 24 2015

nonn

There exists a K-class of Heronian triangles such that the sum of the tangents of their half angles is a constant K > 1, iff K^2-3 is the sum of two squares. E.g., for K = 2 (x=1, y=0) we generate the class of integer Soddyian triangles (see A034017, A210484). For K = 4 (x=2, y=3) the class generated is Heronian triangles with the ratio of r_i : r_o : r = 1 : 3 : 6 where r is their inradius and r_i, r_o are the radii of their inner and outer Soddy circles.
Also because K^2-3 is the sum of two squares it must be congruent to 1 (mod 4). Consequently K is even.
Numbers k such that k^2-3 is in A001481. - Robert Israel, Feb 05 2019
From William P. Orrick, Nov 14 2024: (Start)
Let t = z^2 + z + 1 for some nonnegative integer z, and suppose that t = r * s for some positive integers r and s with r > s. Then (x,y) = (2*z + 1,r - s) has the property that x^2 + y^2 + 3 = (r + s)^2. Hence r + s is a member of this sequence.
Given (x,y) such that x^2 + y^2 + 3 is a square, one of x and y is odd, which we can take to be x, and the other even, which we then take to be y. Let z = (x - 1) / 2. Then 4 * (z^2 + z + 1) + y^2 is an even square, which we can call q^2. Hence z^2 + z + 1 factorizes into integer factors r = (q + y) / 2 and s = (q - y) / 2. Therefore all elements of this sequence are obtained by choosing a nonnegative integer z and a factor r of z^2 + z + 1 and forming the sum r + (z^2 + z + 1) / r.
Using the notation of the preceding two comments, the 2 by 2 matrix [[-z,r],[-s,1+z]] has both determinant and trace equal to 1, implying that it is an element of the modular group of order 3. Forming the product of the order-2 matrix [[0,-1],[1,0]] with this matrix gives the matrix [[s,-1-z],[-z,r]], which has trace r + s. Since all elements of the modular group that are a product of an element of order 2 and an element of order 3 can be obtained from a matrix of the form above by conjugation, this sequence consists of the traces of elements of the modular group that can be expressed as such a product. (This ignores the sign of the trace, which is immaterial if matrices are understood to represent fractional linear transformations.)
(End)

			a(4) = 10 as 10^2 - 3 = 9^2 + 4^2 and 10 is the 4th such occurrence.

Robert Israel, Table of n, a(n) for n = 1..10000
Frank M. Jackson and Stalislav Takhaev, Heronian Triangles of Class K: Congruent Incircles Cevian Perspective, Forum Geom., 15 (2015) 5-12.

Cf. A034017, A210484.

filter:= proc(n) local F;
  F:= ifactors(n^2-3)[2];
  andmap(t -> t[1] mod 4 <> 3 or t[2]::even, F)
end proc:
select(filter, [seq(i,i=2..1000,2)]); # Robert Israel, Feb 05 2019

lst = {}; Do[If[IntegerQ[k=Sqrt[m^2+n^2+3]], AppendTo[lst, k]], {m, 0, 1000}, {n, 0, m}]; Union@lst

from itertools import count, islice
from sympy import factorint
def A250310_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n**2-3).items()),count(2))
A250310_list = list(islice(A250310_gen(),30)) # Chai Wah Wu, Jun 27 2022

Edited by Robert Israel, Feb 05 2019

A364722 Numbers k that divide 1 + 2^m + 4^m for some m.

Original entry on oeis.org

1, 3, 7, 13, 19, 21, 37, 39, 49, 57, 61, 67, 73, 79, 91, 97, 103, 109, 111, 139, 147, 151, 163, 169, 181, 183, 193, 199, 201, 211, 219, 237, 241, 271, 273, 291, 307, 309, 313, 327, 331, 337, 343, 349, 361, 367, 373, 379, 409, 417, 421, 427, 433, 453, 463, 469, 487, 489, 507, 523, 541, 543, 547

Offset: 1

Robert Israel, Aug 04 2023

nonn

If k = 3*j+1 is prime and 2^j - 1 is not divisible by k, then k is a term, as 1 + 2^j + 4^j = (2^(k-1)-1)/(2^j - 1) == 0 (mod k). - Robert Israel, Aug 06 2023

			a(4) = 13 is a term because 1 + 2^4 + 4^4 = 273 = 21 * 13 is divisible by 13.

Robert Israel, Table of n, a(n) for n = 1..10000

Subset of A034017. Cf. A364724.

filter:= proc(n) local x, r;
  for r in map(t -> subs(t,x), [msolve(1+x+x^2, n)]) do
    try
      NumberTheory:-ModularLog(r,2,n);
    catch "no solutions exist": next
    end try;
    return true
  od;
  false
end proc:
select(filter, [seq(i,i=1..1000,2)]);

from itertools import count, islice
from sympy import sqrt_mod_iter, discrete_log
def A364722_gen(startvalue=1): # generator of terms >= startvalue
    if startvalue <= 1:
        yield 1
    if startvalue <= 3:
        yield 3
    for k in count(max(startvalue,4)):
        for d in (r>>1 for r in sqrt_mod_iter(-3,k) if r&1):
            try:
                discrete_log(k,d,2)
            except:
                continue
            yield k
            break
A364722_list = list(islice(A364722_gen(),20)) # Chai Wah Wu, May 02 2024

A034018 Numbers that are represented by x^2+xy+y^2 but not primitively.

Original entry on oeis.org

4, 9, 12, 16, 25, 27, 28, 36, 48, 52, 63, 64, 75, 76, 81, 84, 100, 108, 112, 117, 121, 124, 144, 148, 156, 171, 172, 175, 189, 192, 196, 208, 225, 228, 243, 244, 252, 256, 268, 279, 289, 292, 300, 304, 316, 324, 325, 333, 336, 351, 363, 364, 372, 387, 388, 400

Offset: 1

N. J. A. Sloane

nonn

Cf. A034017.

Extended by Ray Chandler, Jan 29 2009

A034021 Numbers that are primitively but not imprimitively represented by x^2+xy+y^2.

Original entry on oeis.org

0, 1, 3, 7, 13, 19, 21, 31, 37, 39, 43, 57, 61, 67, 73, 79, 91, 93, 97, 103, 109, 111, 127, 129, 133, 139, 151, 157, 163, 181, 183, 193, 199, 201, 211, 217, 219, 223, 229, 237, 241, 247, 259, 271, 273, 277, 283, 291, 301, 307, 309, 313, 327, 331, 337, 349, 367

Offset: 1

N. J. A. Sloane

nonn

Also numbers that are squarefree and primitively represented by x^2+x*y+y^2. - Frank M Jackson, Nov 08 2013

Cf. A034017, A003136, A045897.

lst = {}; Do[k=x^2+x*y+y^2; If[(SquareFreeQ[k] && GCD[x, y]==1) || k==0, AppendTo[lst, k]], {x, 0, 100}, {y, 0, x}]; Union@lst (* Frank M Jackson, Nov 08 2013 *)

Extended by Ray Chandler, Jan 29 2009

A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

Original entry on oeis.org

3, 7, 9, 21, 21, 39, 61, 63, 93, 129, 169, 171, 219, 273, 331, 333, 399, 471, 547, 549, 633, 723, 817, 819, 921, 1029, 1141, 1143, 1263, 1389, 1519, 1521, 1659, 1803, 1951, 1953, 2109, 2271, 2437, 2439, 2613, 2793, 2977, 2979, 3171, 3369, 3571, 3573, 3783, 3999, 4219, 4221, 4449, 4683, 4921, 4923

Offset: 1

Greg Dresden, Jun 25 2018

nonn

After 3, the prime terms appear to be the primes in A275878 (namely, 7, 61, 331, 547, 1951, ...)

			For the first entry of Q=3, we have the polynomial x^3 - 3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are Q=7 with the polynomial x^3 - 7x + 7, then Q=9 with the polynomial x^3 - 9x + 9, and Q=21 with the polynomials x^3 - 21x + 35 and x^3 - 21x + 37. Note that for the Q=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.

Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.

Cf. A316184. Contained in the union of A034017 and three times A034017.

SetOfQRs = {};
M = 1000;
Do[
  If[Divisible[3 (a^2 - a + 1), c^2] &&
    Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&
    3 (a^2 - a + 1)/c^2 <=  M,
   SetOfQRs =
    Union[SetOfQRs, { { (3 (a^2 - a + 1))/
       c^2, ((2 a - 1) (a^2 - a + 1))/c^3}}   ]],
  {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];
Print[SetOfQRs // MatrixForm];

More terms from Robert G. Wilson v, Jul 02 2018

A220171 An ordered subset of primitive values of x^2 + x*y + y^2 where at least two ordered pairs (x1,y1) and (x2,y2) with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1 yield identical primitive values.

Original entry on oeis.org

91, 133, 217, 247, 259, 273, 301, 399, 403, 427, 469, 481, 511, 553, 559, 589, 637, 651, 679, 703, 721, 741, 763, 777, 793, 817, 871, 889, 903, 931, 949, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1183, 1209, 1261, 1267, 1273, 1281, 1333, 1339, 1351

Offset: 1

Frank M Jackson, Dec 06 2012

nonn

The primitive values of x^2 + x*y + y^2 where x >= y >= 0 and gcd(x, y) = 1 are given by A034017. However there are incidents in the sequence A034017 where different values of (x, y) yield the same primitive value. Furthermore, the number of solutions for a given primitive value equates to a power of 2. See A121940.

			a(3) = 217 because it is the 3rd incident in ascending order of the primitive x^2 + x*y + y^2 that yields multiple solutions. This happens when (x, y) = (9, 8) and (13, 3).

Cf. A034017, A121940.

maxLen = 100; sol[k_] := Solve[m^2 + m*n + n^2 == k && m > n > 0 && GCD[m, n] == 1, Integers]; getlist[l_] := Which[Length[sol[l]] == 0, {}, True, {m, n} /. sol[l]]; list = {}; p = 1; While[Length[list] < maxLen, (While[Length[getlist[p]] < 2, p++]; list = Append[list, p]; p++)]; list

n such that n = x1^2 + x1*y1 + y1^2 = x2^2 + x2*y2 + y2^2 with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1.

cp's OEIS Frontend

A250310 Numbers whose squares are of the form x^2 + y^2 + 3 where x >= y >= 0 (repetitions omitted).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A364722 Numbers k that divide 1 + 2^m + 4^m for some m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A034018 Numbers that are represented by x^2+xy+y^2 but not primitively.

Views

Author

Keywords

Crossrefs

Extensions

A034021 Numbers that are primitively but not imprimitively represented by x^2+xy+y^2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A220171 An ordered subset of primitive values of x^2 + x*y + y^2 where at least two ordered pairs (x1,y1) and (x2,y2) with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1 yield identical primitive values.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula