A088977 -id:A088977 - cp's OEIS frontend

A089025 Side of primitive equilateral triangle bearing at least one integral cevian that partitions an edge into two integral sections.

8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112, 117, 119, 133, 143, 153, 160, 168, 171, 176, 187, 207, 209, 221, 224, 225, 247, 253, 255, 264, 275, 280, 285, 299, 312, 319, 323, 325, 341, 345, 352, 360, 377, 391, 403, 408, 416, 425, 435, 437, 440, 448

Offset: 1

Lekraj Beedassy, Nov 12 2003

nonn

The cevians are numbers divisible only by primes of form 6n+1:A002476 (i.e., correspond to entries of A004611).
Composite cevians c belong to more than one equilateral triangle, actually to 2^(omega(c)-1) of them, where omega(n)=A001221(n). For instance, cevian 1813=7^2*37, with omega(1813)=2, belongs to 2^(2-1)=2 equilateral triangles, their sides being 1927=255+1627 and 1960=343+1617, while cevian 1729=7*13*19, with omega(1729)=3, belongs to 2^(3-1)=4 equilateral triangles whose sides are 1775=96+1679, 1824=209+1615, 1840=249+1591, 1859=299+1560.
Given a triangle with integer side lengths a, b, c relatively prime with a < b, c < b, and angle opposite c of 60 degrees then a*a - a*b + b*b = c*c from law of cosines and called a primitive Eisenstein triple by Gordon. This sequence is the possible side lengths of b. - Michael Somos, Apr 11 2012

			The equilateral triangle with side 280, for instance, has cevian 247 partitioning an edge into 93+187, as well as cevian 271 that sections the edge into 19+261.

O. Delgado-Friedrichs and M. O'Keeffe, Edge-transitive lattice nets, Acta Cryst. A, A65 (2009), 360-363.
Russell A. Gordon, Properties of Eisenstein Triples, Mathematics Magazine 85 (2012), 12-25.

Cf. A088513, A088514, A088977.

findPrimIntEquiSide[maxC_] :=
Reap[Do[Do[
     With[{cevian = Abs[c E^((2 \[Pi] I)/6) - a]},
      If[FractionalPart[cevian] == 0 && GCD[a, c] == 1,
       Sow[c]; Break[]]], {a, Floor[c/2],
      1, -1}], {c, maxC}]][[2, 1]]
(* Andrew Turner, Aug 04 2017 *)

A088241 Values of y, where x^2 + xy + y^2 = p (xA002476).

Original entry on oeis.org

2, 3, 3, 5, 4, 6, 5, 7, 8, 7, 8, 9, 7, 7, 10, 9, 12, 11, 11, 9, 13, 14, 11, 12, 15, 10, 12, 13, 17, 16, 11, 13, 17, 13, 17, 15, 12, 15, 20, 13, 18, 17, 21, 21, 18, 17, 21, 14, 21, 19, 24, 23, 19, 22, 15, 18, 20, 21, 19, 25, 18, 19, 23, 21, 27, 17, 27, 25, 19, 20, 27, 23, 28, 21, 26

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

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

Cf. A002476, A088242, A088243, A088296, A088298, A088299, A088977.

R:= NULL: count:= 0:
for k from 1 while count < 100 do
  p:= 6*k+1;
  if not isprime(p) then next fi;
  S:= select(t -> subs(t,x) > 0 and subs(t,x) < subs(t,y), [isolve(x^2+x*y+y^2=p)]);
  S:= map(t -> subs(t,y), S);
  R:= R,op(S); count:= count+1;
od:
R; #Robert Israel, Jun 16 2025

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088242 Values of x, where x^2 + xy + y^2 = p (xA002476).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 1, 3, 3, 2, 5, 6, 3, 5, 1, 3, 4, 7, 2, 1, 6, 5, 1, 9, 7, 6, 1, 3, 10, 8, 3, 9, 4, 7, 11, 8, 1, 11, 5, 7, 1, 2, 7, 9, 4, 13, 5, 8, 1, 3, 9, 5, 14, 11, 9, 8, 11, 3, 13, 12, 7, 10, 1, 15, 2, 6, 14, 13, 4, 10, 3, 13, 7, 17, 3, 7, 9, 13, 8, 11, 16, 15, 6, 3, 12, 17, 7, 9, 1, 3, 16

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

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

Cf. A002476, A088241, A088243, A088296, A088298, A088299, A088977.

R:= NULL: count:= 0:
for k from 1 while count < 100 do
  p:= 6*k+1;
  if not isprime(p) then next fi;
  S:= select(t -> subs(t,x) > 0 and subs(t,x) < subs(t,y), [isolve(x^2+x*y+y^2=p)]);
  S:= map(t -> subs(t,x), S);
   R:= R,op(S); count:= count+1;
od:
R; # Robert Israel, Jun 16 2025

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[x /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088243 Values of x + y, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

3, 4, 5, 6, 7, 7, 9, 9, 9, 10, 11, 11, 12, 13, 13, 14, 13, 14, 15, 16, 15, 15, 17, 17, 16, 19, 19, 19, 18, 19, 21, 21, 20, 22, 21, 22, 23, 23, 21, 24, 23, 24, 22, 23, 25, 26, 25, 27, 26, 27, 25, 26, 28, 27, 29, 29, 29, 29, 30, 28, 31, 31, 30, 31, 28, 32, 29, 31, 33, 33, 31, 33

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088296, A088298, A088299, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[x + y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088296 Values of y - x, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

1, 2, 1, 4, 1, 5, 1, 5, 7, 4, 5, 7, 2, 1, 7, 4, 11, 8, 7, 2, 11, 13, 5, 7, 14, 1, 5, 7, 16, 13, 1, 5, 14, 4, 13, 8, 1, 7, 19, 2, 13, 10, 20, 19, 11, 8, 17, 1, 16, 11, 23, 20, 10, 17, 1, 7, 11, 13, 8, 22, 5, 7, 16, 11, 26, 2, 25, 19, 5, 7, 23, 13, 25, 8, 19, 1, 26, 20, 17, 10, 19, 14, 5, 7, 23

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088243, A088298, A088299, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers];
Sow[y - x /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 09 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088298 Values of 2x + y, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

4, 5, 7, 7, 10, 8, 13, 11, 10, 13, 14, 13, 17, 19, 16, 19, 14, 17, 19, 23, 17, 16, 23, 22, 17, 28, 26, 25, 19, 22, 31, 29, 23, 31, 25, 29, 34, 31, 22, 35, 28, 31, 23, 25, 32, 35, 29, 40, 31, 35, 26, 29, 37, 32, 43, 40, 38, 37, 41, 31, 44, 43, 37, 41, 29, 47, 31, 37, 47, 46, 35

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088243, A088296, A088299, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers];
Sow[2x + y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 09 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088299 Values of x + 2y, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

5, 7, 8, 11, 11, 13, 14, 16, 17, 17, 19, 20, 19, 20, 23, 23, 25, 25, 26, 25, 28, 29, 28, 29, 31, 29, 31, 32, 35, 35, 32, 34, 37, 35, 38, 37, 35, 38, 41, 37, 41, 41, 43, 44, 43, 43, 46, 41, 47, 46, 49, 49, 47, 49, 44, 47, 49, 50, 49, 53, 49, 50, 53, 52, 55, 49, 56, 56, 52, 53

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088243, A088296, A088298, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers];
Sow[x + 2y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 09 2020 *)

More terms from Ray Chandler, Nov 04 2003

A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

Original entry on oeis.org

341880, 8168160, 14636160, 17957940, 52492440, 116396280, 1071572040, 1187525640, 1728483120, 5988702720, 6609482880, 22539095040, 29239970760, 136496680320, 258670630680, 398648544840, 494892478080, 592003418160, 1329673884000, 1343798407560, 2190884461920

Offset: 1

Sture Sjöstedt, Aug 23 2017

nonn

Numbers n such that there exist positive integers x, y with x > y and n = x*y*(x-y)*(x+y).

Many of them consist of a Pythagorean triangle plus a triple which is a solution to Carroll's problem: Find three Pythagorean triangles with the same area.

			p^2 - p*q + q^2 = r^2;
p = 208, q = 418, r = 362, q - p = 210;
n = p*r*q*(q-p) = 208*418*362*210 = 6609482880.
x = 640, y = 627 gives the same area:
n = x*y*(x-y)*(x+y) = 640*627*13*1267 = 6609482880.

Cf. A009127, A024407, A055193, A088513, A088977, A089025, A177021, A291591.

a(12)-a(21) from Giovanni Resta, Aug 28 2017

cp's OEIS Frontend

A089025 Side of primitive equilateral triangle bearing at least one integral cevian that partitions an edge into two integral sections.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A088241 Values of y, where x^2 + xy + y^2 = p (xA002476).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A088242 Values of x, where x^2 + xy + y^2 = p (xA002476).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A088243 Values of x + y, where x^2 + xy + y^2=p (xA002476).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A088296 Values of y - x, where x^2 + xy + y^2=p (xA002476).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A088298 Values of 2x + y, where x^2 + xy + y^2=p (xA002476).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A088299 Values of x + 2y, where x^2 + xy + y^2=p (xA002476).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions