A089025 -id:A089025 - cp's OEIS frontend

A088513 Smallest cevians corresponding to A089025(n).

7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 91, 97, 103, 109, 127, 133, 133, 139, 157, 151, 169, 163, 193, 181, 199, 211, 217, 217, 223, 241, 229, 247, 247, 259, 259, 277, 301, 307, 283, 331, 301, 313, 343, 337, 349, 367, 397, 361, 373, 379, 403, 421, 403

Offset: 1

Lekraj Beedassy, Nov 14 2003

nonn

Cf. A089025.

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

A088514 Smaller section of edge A089025(n) by integral cevian A088513(n).

Original entry on oeis.org

3, 7, 5, 11, 7, 13, 16, 9, 32, 17, 40, 11, 19, 55, 40, 24, 13, 23, 65, 69, 25, 56, 15, 75, 32, 104, 56, 29, 17, 87, 85, 31, 119, 72, 93, 64, 144, 95, 40, 35, 133, 21, 136, 105, 37, 105, 111, 88, 23, 185, 152, 176, 80, 41, 115

Offset: 1

Lekraj Beedassy, Nov 14 2003

nonn

Obviously, larger section is A089025(n) - a(n).

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

Larger section is A089025(n) - a(n) = A088586(n). a(n) = (A089025(n) - A088587(n))/2. - Lekraj Beedassy, Nov 25 2003

a(7) changed from 18 to 16 by W. D. Smith, Mar 04 2012 (N. J. A. Sloane, Dec 05 2012)

a(42) changed from 105 to 21, and a(52)=176 inserted by Andrew Turner, Aug 04 2017

A088586 Larger section of edge A089025(n) by integral cevian A088513(n).

Original entry on oeis.org

5, 8, 16, 24, 33, 35, 39, 56, 45, 63, 51, 85, 80, 57, 77, 95, 120, 120, 88, 91, 143, 115, 161, 112, 175, 105, 165, 195, 208, 160, 168, 224, 145, 203, 187, 221, 155, 217, 279, 288, 192, 320, 209, 247, 323, 272, 280, 315, 385, 231, 273, 259, 357, 399, 333

Offset: 1

Lekraj Beedassy, Nov 20 2003

nonn

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

a(n) = (A089025(n) + A088587(n))/2. - Lekraj Beedassy, Nov 25 2003

Repeated 112 at a(24) and a(25) removed, and order of a(35)=187 and a(36)=221 reversed by Andrew Turner, Aug 04 2017

A088587 Difference between the two segments of edge d = A089025(n) partitioned by cevian c = A088513(n).

Original entry on oeis.org

2, 1, 11, 13, 26, 22, 23, 47, 13, 46, 11, 74, 61, 2, 37, 71, 107, 97, 23, 22, 118, 59, 146, 37, 143, 1, 109, 166, 191, 73, 83, 193, 26, 131, 94, 157, 11, 122, 239, 253, 59, 299, 73, 142, 286, 167, 169, 227, 362, 46, 121, 83, 277, 358, 218

Offset: 1

Lekraj Beedassy, Nov 20 2003

nonn

a(n) is minimal, i.e., equals 1 for values c = A001570(n), d = A028230(n).

Cf. A088514.

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

a(n) = sqrt(4c^2 - 3d^2).

a(35)=94 and a(36)=157 order reversed by Andrew Turner, Aug 04 2017

A335896 Largest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

Original entry on oeis.org

8, 8, 15, 15, 21, 21, 35, 35, 40, 40, 48, 48, 55, 55, 65, 65, 77, 77, 80, 80, 91, 91, 96, 96, 99, 99, 112, 112, 117, 117, 119, 119, 133, 133, 143, 143, 153, 153, 160, 160, 171, 171, 168, 168, 187, 187, 176, 176, 209, 209, 207, 207, 221, 221, 224, 224, 225, 225

Offset: 1

Bernard Schott, Jul 10 2020

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of middle side, and if middle sides coincide then by increasing order of the largest side. This sequence lists the c's.
Equivalently, lengths of the largest side c of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to the largest angle C.
Also, solutions c of the Diophantine equation b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.
For the corresponding primitive triples and miscellaneous properties and references, see A335893.
When (a, b, c) is a triple, then (c-a, b, c) is another triple, so every c in the data is twice consecutively present according to the corresponding pair (b, c) (see examples).
As B = Pi/3 and C runs from Pi/3 to 2*Pi/3, sin(C) gets a maximum when C = Pi/2 with sin(C) = 1, hence, from law of sinus, b/sin(B) = c/sin(C), c < b/sin(Pi/3) = b * 2/sqrt(3) < 6*b/5. This bound is used in PARI and Maple programs.
This sequence is not increasing. For example, a(41) = a(42) = 171 for triangle with middle side = 151 while a(43) = a(44) = 168 for triangle with middle side = 157.

			c = 8 appears twice because:
  7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
  7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
c = 96 and c = 99 each appear twice associated with b = 91 because:
  91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
  91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
  91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
  91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.

Wikipedia, Law of sines.

Cf. A089025 (terms in increasing order without repetition).

Cf. A335893 (triples), A335894 (smallest side), A335895 (middle side), this sequence (largest side), A335897 (perimeter).

for b from 3 to 250 by 2 do
for c from b+1 to 6*b/5 do
a := (c - sqrt(4*b^2-3*c^2))/2;
if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(c,c); end if;
end do;
end do;

lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print(c, ", ", c, ", ");););););} \\ Michel Marcus, Jul 15 2020

a(n) = A335893(n, 3).

c satisfies c^2 - a*c + a^2 - b^2 = 0 with gcd(a,b) = 1 and a < b.

A061281 Side of n-th equilateral triangle enclosing at least one point located at integer distances from the vertices.

Original entry on oeis.org

112, 147, 185, 224, 273, 283, 294, 331, 331, 336, 370, 403, 441, 448, 485, 520, 546, 555, 559, 560, 566, 588, 592, 637, 645, 662, 662, 672, 691, 735, 740, 784, 806, 819, 849, 882, 896, 925, 965, 970, 993, 993, 1008, 1029, 1040, 1047, 1092, 1110, 1118, 1120, 1132

Offset: 1

Lekraj Beedassy, May 21 2001

nonn

The equation has many other integer solutions, such as {3,5,7,8}; most of these describe points that lie on the edge of the triangle. - David Wasserman, Jun 10 2002. See A089025.

			The solution (97,185,208,273) of the triangle equation gives rise to the value 273 as the 5th equilateral triangle associated with an interior point at integer distances from the vertices.

M. Gardner, Mathematical Circus, Alfred A. Knopf, 1979, p. 65.
L. Pianaro, Pierre Est Encore Perdu, Jouer Jeux Mathematiques, No. 18, Oct 1995, published by French Federation of Mathematics Games.

Cf. A072052, A072053, A072054, A089025.

a(n) is the largest term in the n-th quadruple (a, b, c, d) satisfying the triangle equation 3*(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.

More terms from David Wasserman, Jun 10 2002

More terms from Jinyuan Wang, Jul 20 2020

A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.

Original entry on oeis.org

3, 7, 8, 5, 7, 8, 5, 19, 21, 7, 13, 15, 7, 37, 40, 8, 13, 15, 9, 61, 65, 11, 31, 35, 11, 91, 96, 13, 43, 48, 13, 127, 133, 15, 169, 176, 16, 19, 21, 16, 49, 55, 17, 73, 80, 17, 217, 225, 19, 91, 99, 19, 271, 280, 21, 331, 341, 23, 133, 143, 23, 397, 408

Offset: 1

Colin Barker, Nov 26 2015

The sides of a primitive 60-degree integer triangle.

Colin Barker, Table of n, a(n) for n = 1..9999
Eric Weisstein's World of Mathematics, Pythagorean Triple
Wikipedia, Eisenstein triple

Cf. A089025, A121992, A201223, A263728, A264827.

pt60(a) = {
  my(L=List(), n=-3*a^2, f, g, b, c);
  fordiv(n, f,
    g=n\f;
    if(f>g && (g+f)%2==0 && (f-g)%4==0,
      b=(f-g)\4; c=((f+g)\2+a)\2;
      if(c>0 && a

A229838 Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105

Offset: 1

Colin Barker, Oct 01 2013

nonn

A primitive triangle is one for which the sides have no common factor.

A004611 gives the values of B, and A089025 gives the values of C.

			7 appears in the sequence because there exists a primitive 60-degree triangle with sides 7, 37 and 40.

Wikipedia, Integer triangle

Cf. A004611, A089025, A229839.

\\ Gives terms not exceeding amax
\\ e.g. pt60a(25) gives [3,5,7,8,9,11,13,15,16,17,19,21,23,24,25]
pt60a(amax) = {
  s=[];
  for(m=1, amax\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        if(2*m*n+n*n<=amax, s=concat(s, 2*m*n+n*n));
        if(m*m-n*n<=amax, s=concat(s, m*m-n*n))
      )
    )
  );
  vecsort(s,,8)
}

Empirical g.f.: -x*(x^5-x^4-x^3-2*x^2-2*x-3) / ((x-1)^2*(x^4+x^3+x^2+x+1)).

A242039 List of integers b such that (a1,b,c1) and (a2,b,c2) are primitive Eisenstein triples, max(a1,b,c1,a2,c2)=b, and a1,c1,a3,c3 are distinct.

Original entry on oeis.org

280, 1144, 1155, 1680, 1768, 1976, 2145, 2584, 2805, 3003, 3128, 3315, 3360, 3400, 3496, 3705, 3800, 4095, 4600, 4845, 5005, 5280, 5336, 5355, 5704, 5720, 5800, 5865, 5985, 6160, 6200, 6240, 6545, 6555, 6783, 6864, 7192, 7280, 7315, 7400, 7735, 8120, 8265, 8584, 8645, 8680, 8835, 8855, 9176, 9177, 9240, 9360, 9512, 9976

Offset: 1

Albert Lau, Aug 12 2014

nonn

For Eisenstein triple see A121992.

			280 is in the list because (93,280,247) and (19,280,271) are primitive Eisenstein triples and 280 is the largest side and no other side is equal.
Consider (3,8,7) and (5,8,7), 8 is not in the list because 7 appear in both triple.

Cf. A089025, A121992.

max = 2000;
data = Do[Sqrt[-3 a^2 + 4 c^2] // If[IntegerQ[#] && GCD[a, c] == 1, {a, (a + #)/2, c} // Sow] &, {a, max}, {c, Sqrt[3]/2 a // Ceiling, a - 1}] // Reap // Last // Last;
Select[data[[;; , 1]] // Tally, #[[2]] > 1 &][[;; , 1]]

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

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A088513 Smallest cevians corresponding to A089025(n).

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A088514 Smaller section of edge A089025(n) by integral cevian A088513(n).

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A088586 Larger section of edge A089025(n) by integral cevian A088513(n).

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Mathematica

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A088587 Difference between the two segments of edge d = A089025(n) partitioned by cevian c = A088513(n).

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A335896 Largest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

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Maple

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A061281 Side of n-th equilateral triangle enclosing at least one point located at integer distances from the vertices.

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Extensions

A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.

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PARI

A229838 Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.

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Examples

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Programs

PARI

Formula

A242039 List of integers b such that (a1,b,c1) and (a2,b,c2) are primitive Eisenstein triples, max(a1,b,c1,a2,c2)=b, and a1,c1,a3,c3 are distinct.

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