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 1-4 of 4 results.

A353619 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

Original entry on oeis.org

3, 35, 119, 112, 279, 20, 253, 539, 552, 91, 923, 533, 476, 1455, 224, 1504, 17, 799, 2159, 1513, 1476, 437, 1387, 3059, 2261, 1240, 3160, 4179, 2163, 748, 3212, 391, 1817, 5543, 3151, 4393, 5712, 1175, 2825, 7175, 5825, 2548, 5876, 189, 9099, 4077, 5859, 1736, 9352, 5768, 1189
Offset: 1

Views

Author

Bernard Schott, May 07 2022

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c; hence, this sequence of sides a is not increasing.
In the case B = 3*C, the corresponding metric relation between sides is c*a^2= (b-c)^2 * (b+c).
Equivalently, length of side common to the two angles, one being the triple of the other, of a primitive integer-sided triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the smallest side a = 3 of the first triple (3, 10, 8).
c < a < b with the middle side a = 35 of the 2nd triple (35, 48, 27).
c < b < a with the largest side a = 539 of the 8th triple (539, 510, 216), the first of this type.

Links

  • The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Crossrefs

Cf. A353618 (triples), this sequence (side a), A353620 (side b), A353621 (side c), A353622 (perimeter).
Cf. A343064 (similar, but with B = 2*C).

Programs

  • Maple
    for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a,b,q)=1 and (b-q^3) < a and a < b+q^3 then print(a); end if;
end do;
end do;

Formula

a(n) = A353618(n, 1).

A353620 Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

Original entry on oeis.org

10, 48, 132, 195, 280, 357, 504, 510, 665, 792, 840, 840, 1035, 1288, 1485, 1575, 1740, 1848, 1872, 1890, 2233, 2496, 2604, 2610, 2640, 3003, 3069, 3520, 3536, 3885, 4095, 4368, 4560, 4620, 4662, 4680, 5291, 5712, 5904, 5928, 6006, 6579, 6765, 6992, 7462, 7480, 7568, 8037, 8385, 8415, 8820
Offset: 1

Views

Author

Bernard Schott, May 07 2022

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of side c.
In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
Equivalently, length of side opposite to the angle that is the triple of another one, for primitive integer-sided triangles.
Note that side b is never the smallest side of the triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the largest side b = 10 of the first triple (3, 10, 8).
c < a < b with the largest side b = 48 of the 2nd triple (35, 48, 27).
c < b < a with the middle side b = 510 of the 8th triple (539, 510, 216), the first of this type.
The first side b for which there exist two distinct triangles with B = 3*C is for a(11) = a(12) = 840, and these sides b belong respectively to triples (923, 840, 343) and (533, 840, 512).

Links

  • The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Crossrefs

Cf. A353618 (triples), A353619 (side a), this sequence (side b), A353621 (side c), A353622 (perimeter).
Cf. A343065 (similar, but with B = 2*C).

Programs

  • Maple
    for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a,b,q)=1 and (b-q^3) < a and a < b+q^3 then print(b); end if;
end do;
end do;

Formula

a(n) = A353618(n, 2).

A353621 Side c of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

Original entry on oeis.org

8, 27, 64, 125, 125, 343, 343, 216, 343, 729, 343, 512, 729, 512, 1331, 729, 1728, 1331, 729, 1000, 1331, 2197, 1728, 1000, 1331, 2197, 1331, 1331, 2197, 3375, 2197, 4096, 3375, 1728, 2744, 2197, 2197, 4913, 4096, 2197, 2744, 4913, 3375, 6859, 2744, 4913, 4096, 6859, 3375, 4913, 8000
Offset: 1

Views

Author

Bernard Schott, May 07 2022

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of side c; hence, this sequence of sides c is not increasing.
In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
Equivalently, length of side opposite to the angle that is one-third of another one, for primitive integer-sided triangles.
All terms are cubes >= 8 (A000578). More generally, when c is the side of a primitive integer-sided triangles (a, b, c) whose angle B = m*C, then c = k^m, for some k >= 2.
Note that side c is never the largest side of the triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the middle side c = 8 of the first triple (3, 10, 8).
c < a < b with the smallest side c = 27 of the 2nd triple (35, 48, 27).
c < b < a with the smallest side c = 216 of the 8th triple (539, 510, 216), the first of this type.
The smallest side c for which there exist two distinct triangles with B = 3*C is for a(4) = a(5) = 125, and these sides c belong respectively to triples (112, 195, 125) and (279, 280, 125).

Links

  • The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Crossrefs

Cf. A353618 (triples), A353619 (side a), A353620 (side b), this sequence (side c), A353622 (perimeter).
Cf. A343066 (similar, but with B = 2*C).
Cf. A000578.

Programs

  • Maple
    for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a,b,q)=1 and (b-q^3) < a and a < b+q^3 then print(q^3); end if;
end do;
end do;

Formula

a(n) = A353618(n, 3).

A353622 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 3*C.

Original entry on oeis.org

21, 110, 315, 432, 684, 720, 1100, 1265, 1560, 1612, 2106, 1885, 2240, 3255, 3040, 3808, 3485, 3978, 4760, 4403, 5040, 5130, 5719, 6669, 6232, 6440, 7560, 9030, 7896, 8008, 9504, 8855, 9752, 11891, 10557, 11270, 13200, 11800, 12825, 15300, 14575, 14040, 16016
Offset: 1

Views

Author

Bernard Schott, May 08 2022

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c.
This sequence is nonincreasing: a(12) = 1885 < a(11) = 2106.
If in triangle ABC, B = 3*C, then the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 21 with a < c < b for the first triple (3, 10, 8);
a(2) = 110 with c < a < b for the 2nd triple (35, 48, 27);
a(8) = 1265 with c < b < a for the 8th triple (539, 510, 216).

Links

  • The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Crossrefs

Cf. A353618 (triples), A353619 (side a), A353620 (side b), A353621 (side c).
Cf. A343067 (similar, but with B = 2*C).

Programs

  • Maple
    for b from 4 to 9000 do
  for q from 2 to floor((b-1)^(1/3)) do
    a := (b-q^3) * sqrt(1+b/q^3);
    if a=floor(a) and q^3 < b and igcd(a,b,q)=1 and b-q^3 < a and a < b+q^3 then
      print(a+b+q^3);
    end if;
  end do;
end do;
  • PARI
    lista(nn) = {for (b = 4 , nn, for (q = 2 , sqrtnint(b-1, 3), if (issquare(z=1+b/q^3), a = (b-q^3) * sqrtint(numerator(z))/sqrtint(denominator(z)); if ((q^3 < b) && (gcd([a, b, q])==1) && (b-q^3 < a) && (a < b+q^3), print1(a+b+q^3, ", ");););););} \\ Michel Marcus, May 11 2022

Formula

a(n) = A353618(n, 1) + A353618(n, 2) + A353618(n, 3).
a(n) = A353619(n) + A353620(n) + A353621(n).
Showing 1-4 of 4 results.

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