A339857 -id:A339857 - cp's OEIS frontend

A339856 Primitive triples for integer-sided triangles whose sides a < b < c form a geometric progression.

4, 6, 9, 9, 12, 16, 16, 20, 25, 25, 30, 36, 25, 35, 49, 25, 40, 64, 36, 42, 49, 49, 56, 64, 49, 63, 81, 49, 70, 100, 49, 77, 121, 64, 72, 81, 64, 88, 121, 81, 90, 100, 81, 99, 121, 81, 117, 169, 81, 126, 196, 100, 110, 121, 100, 130, 169, 121, 132, 144, 121, 143, 169

Offset: 1

Bernard Schott, Dec 19 2020

These triangles are called "geometric triangles" in Project Euler problem 370 (see link).
The triples are displayed in increasing lexicographic order (a, b, c).
Equivalently: triples of integer-sided triangles such that b^2 = a*c with a < c and gcd(a, c) = 1.
When a < b < c are in geometric progression with b = a*q, c = b*q, q is the constant, then 1 < q < (1+sqrt(5))/2 = phi = A001622 = 1.6180... (this bound is used in Maple code).
For each triple (a, b, c), there exists (r, s), 0 < r < s such that a = r^2, b = r*s, c = s^2, q = s/r.
Angle C < 90 degrees if 1 < q < sqrt(phi) and angle C > 90 degrees if sqrt(phi) < q < phi with sqrt(phi) = A139339 = 1.2720...
For k >= 2, each triple (a, b, c) of the form (k^2, k*(k+1), (k+1)^2) is (A008133(3k+1), A008133(3k+2), A008133(3k+3)).
Three geometrical properties about these triangles:
  1) The sinus satisfy sin^2(B) = sin(A) * sin(C) with sin(A) < sin(B) < sin(C) that form a geometric progression.
  2) The heights satisfy h_b^2 = h_a * h_c with h_c < h_b < h_a that form a geometric progression.
  3) b^2 = 2 * R * h_b, with R = circumradius of the triangle ABC.

			The smallest such triangle is (4, 6, 9) with 4*9 = 6^2.
There exist four triangles with small side = 49 corresponding to triples (49, 56, 64), (49, 63, 81), (49, 70, 100) and (49, 77, 121).
The table begins:
   4,  6,  9;
   9, 12, 16;
  16, 20, 25;
  25, 30, 36;
  25, 35, 49;
  25, 40, 64;
  36, 42, 49;
  ...

David A. Corneth, Table of n, a(n) for n = 1..10002
Project Euler, Problem 370: Geometric Triangles.

Cf. A339857 (smallest side), A339858 (middle side), A339859 (largest side), A339860 (perimeter).
Cf. A336755 (similar for sides in arithmetic progression).
Cf. A335893 (similar for angles in arithmetic progression).
Cf. A001622 (phi), A139339 (sqrt(phi)), A008133.

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 * a) do
for c from b+1 to floor((1+sqrt(5))/2 * b) do
k:=a*c;
if k=b^2 and igcd(a,b,c)=1 then print(a,b,c); end if;
end do;
end do;
end do;

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a,b,c])==1), print([a,b,c])););););} \\ Michel Marcus, Dec 25 2020

upto(n) = my(res=List(), phi = (sqrt(5)+1) / 2); for(i = 2, sqrtint(n), for(j = i+1, (i*phi)\1, if(gcd(i, j)==1, listput(res, [i^2, i*j, j^2])))); concat(Vec(res)) \\ David A. Corneth, Dec 25 2020

Data corrected by David A. Corneth, Dec 25 2020

A339858 Middle side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

Original entry on oeis.org

6, 12, 20, 30, 35, 40, 42, 56, 63, 70, 77, 72, 88, 90, 99, 117, 126, 110, 130, 132, 143, 154, 165, 176, 187, 156, 204, 228, 182, 195, 208, 221, 234, 247, 260, 273, 210, 238, 266, 240, 255, 285, 330, 345, 272, 304, 336, 368, 400, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459

Offset: 1

Bernard Schott, Dec 29 2020

nonn

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order. This sequence lists the b's.
For the corresponding primitive triples and miscellaneous properties and references, see A339856.
This sequence is not increasing. For example, a(11) = 77 for triple (49, 77, 121) while a(12) = 72 for triple (64, 72, 81).
Oblong numbers k*(k+1) >= 6 form a subsequence (A002378) and belong to triples of the form (k^2, k*(k+1), (k+1)^2).

			a(1) = 6 only for the smallest such triangle (4, 6, 9) with 6^2 = 4*9 and a ratio q = 3/2.
a(2) = 12 only for the triangle (9, 12, 16) with 12^2 = 9*16 and a ratio q = 4/3.

Project Euler, Problem 370: Geometric triangles.

Cf. A339856 (triples), A339857 (smallest side), this sequence (middle side), A339859 (largest side), A339860 (perimeter).
Cf. A336751 (similar for sides in arithmetic progression).
Cf. A335894 (similar for angles in arithmetic progression).
Cf. A002378 \ {0,2} (a subsequence).

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 *a) do
for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(b); end if;
end do;
end do;
end do;

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print1(b, ", "); ); ); ); );} \\ Michel Marcus, Dec 30 2020

a(n) = A339856 (n, 2).

A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

Original entry on oeis.org

9, 16, 25, 36, 49, 64, 49, 64, 81, 100, 121, 81, 121, 100, 121, 169, 196, 121, 169, 144, 169, 196, 225, 256, 289, 169, 289, 361, 196, 225, 256, 289, 324, 361, 400, 441, 225, 289, 361, 256, 289, 361, 484, 529, 289, 361, 441, 529, 625, 324, 361, 400, 441, 484, 529, 576

Offset: 1

Bernard Schott, Jan 05 2021

nonn

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order. This sequence lists the c's.
For the corresponding primitive triples and miscellaneous properties and references, see A339856.
The terms are all squares >= 9 but they are not in increasing order. For example, a(6) = 64 for triple (25, 40, 64) while a(7) = 49 for triple (36, 42, 49).

			a(1) = 9 for only the smallest such triangle (4, 6, 9) with 6^2 = 4*9, this one corresponds to an obtuse triangle because sqrt(phi) < q = 3/2 < phi, hence C > Pi/2.
a(3) = 25 for only the triple (16, 20, 25) with 16 * 25 = 20^2, this one corresponds to an acute triangle because 1 < q = 5/4 < sqrt(phi), hence C < Pi/2.

Project Euler, Problem 370: Geometric triangles.

Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), this sequence (largest side), A339860 (perimeter).
Cf. A336753 (similar for sides in arithmetic progression).
Cf. A335896 (similar for angles in arithmetic progression).

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 *a) do
for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(c); end if;
end do;
end do;
end do;

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print1(c, ", "); ); ); ); ); } \\ Michel Marcus, Jan 07 2021

a(n) = A339856(n, 3).

A339860 Perimeter of primitive integer-sided triangles whose sides a < b < c form a geometric progression.

Original entry on oeis.org

19, 37, 61, 91, 109, 129, 127, 169, 193, 219, 247, 217, 273, 271, 301, 367, 403, 331, 399, 397, 433, 471, 511, 553, 597, 469, 637, 733, 547, 589, 633, 679, 727, 777, 829, 883, 631, 723, 823, 721, 769, 871, 1039, 1099, 817, 921, 1033, 1153, 1281, 919, 973, 1029, 1087

Offset: 1

Bernard Schott, Jan 08 2021

nonn

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order.
These perimeters are of the form r^2 + r*s + s^2, r < s, gcd(r, s) = 1 and q = r/s (A034017), so they are all odd but not in increasing order. For example, a(6) = 129 for triple (25, 40, 64) while a(7) = 127 for triple (36, 42, 49).
For the corresponding primitive triples and miscellaneous properties, see A339859.

			a(1) = 19 = 4+6+9 for the smallest such triangle (4, 6, 9) with 4 * 9 = 6^2 and a ratio q = 3/2.
a(2) = 37 = 9+12+16 for the triple (9, 12, 16) with 9 * 16 = 12^2 and a ratio q = 4/3.

Project Euler, Problem 370: Geometric triangles.

Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), A339859 (largest side), this sequence (perimeter).
Cf. A336754 (similar for sides in arithmetic progression).
Cf. A335897 (similar for angles in arithmetic progression).
Subsequence of A034017.

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 *a) do
for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(a+b+c); end if;
end do;
end do;
end do;

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print1(a+b+c, ", ")); ); ); ); } \\ Michel Marcus, Jan 08 2021

a(n) = A339856(n, 1) + A339856(n, 2) + A339856(n, 3).

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A339856 Primitive triples for integer-sided triangles whose sides a < b < c form a geometric progression.

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A339858 Middle side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

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A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

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A339860 Perimeter of primitive integer-sided triangles whose sides a < b < c form a geometric progression.

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