cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A088977 Side of primitive equilateral triangle with prime cevian p=A002476(n) cutting an edge into two integral parts.

Original entry on oeis.org

8, 15, 21, 35, 40, 48, 65, 77, 80, 91, 112, 117, 119, 133, 160, 168, 171, 187, 207, 209, 221, 224, 253, 255, 264, 280, 312, 323, 325, 341, 352, 377, 391, 403, 408, 425, 435, 440, 455, 465, 483, 504, 525, 527, 560, 576, 595, 609, 624, 645, 651, 665, 667, 703
Offset: 1

Views

Author

Lekraj Beedassy, Oct 31 2003

Keywords

Comments

The edge a(n) is partitioned into q=s^2 - t^2=A088243(n)*A088296(n) and r=t(2s+t)=A088242(n)*A088299(n) by a cevian of length p. [Alternatively, (p,q,r) form a triangle with angle 2pi/3 opposite side p.] The quadruple {p,q,r,a(n)=q+r} satisfies the triangle relation: see A061281, or the simpler relation a(n)^2 = p^2 + q*r.

Links

Crossrefs

Cf. A002476, A088241, A088242, A088243, A088296, A088298, A088299.

Programs

  • Mathematica
    sol[p_] := Solve[0 < t < s && s^2 + s t + t^2 == p, {s, t}, Integers];
Union[Reap[For[n = 1, n <= 10000, n++, If[PrimeQ[p = 6n + 1], an = s(s + 2t) /. sol[p][[1]]]; Sow[an]]][[2, 1]]] (* Jean-François Alcover, Mar 06 2020 *)

Formula

a(n) = A088241(n)*A088298(n) = s(s+2t), where s^2 + st + t^2, with s>t, form the primes p = 1 (mod 6) = A002476(n).

Extensions

More terms from Ray Chandler, Nov 01 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

Views

Author

Lekraj Beedassy, Nov 03 2003

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Extensions

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

Views

Author

Lekraj Beedassy, Nov 03 2003

Keywords

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Lekraj Beedassy, Nov 03 2003

Keywords

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Lekraj Beedassy, Nov 03 2003

Keywords

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Lekraj Beedassy, Nov 03 2003

Keywords

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

More terms from Ray Chandler, Nov 04 2003
Showing 1-6 of 6 results.

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