A343894 -id:A343894 - cp's OEIS frontend

A020883 Ordered long legs of primitive Pythagorean triangles.

4, 12, 15, 21, 24, 35, 40, 45, 55, 56, 60, 63, 72, 77, 80, 84, 91, 99, 105, 112, 117, 120, 132, 140, 143, 144, 153, 156, 165, 168, 171, 176, 180, 187, 195, 208, 209, 220, 221, 224, 231, 240, 247, 252, 253, 255, 260, 264, 272, 273, 275, 285, 288, 299, 304, 308, 312, 323

Offset: 1

Clark Kimberling

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A < B); sequence gives values of B, sorted.
Any term in this sequence is given by f(m,n) = 2*m*n or g(m,n) = m^2 - n^2 where m and n are any two positive integers, m > 1, n < m, the greatest common divisor of m and n is 1, m and n are not both odd; e.g., f(m,n) = f(2,1) = 2*2*1 = 4. - Agola Kisira Odero, Apr 29 2016
All terms are composite. - Thomas Ordowski, Mar 12 2017
a(1) is the only power of 2. - Torlach Rush, Nov 08 2019
The first term appearing twice is 420 = a(75) = a(76) = A024410(1). - Giovanni Resta, Nov 11 2019
From Bernard Schott, May 05 2021: (Start)
Also, ordered sides a of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.
Example: a(2) = 12, because the second triple is (12, 10, 15) with side a = 12, satisfying 2/12 = 1/10 + 1/15 and 15-12 < 10 < 15+12.
The first term appearing twice 420 corresponds to triples (420, 310, 651) and (420, 406, 435), the second one is 572 = a(101) = a(102) = A024410(2) and corresponds to triples (572, 407, 962) and (572, 455, 770). The terms that appear more than once in this sequence are in A024410.
For the corresponding primitive triples and miscellaneous properties and references, see A343891. (End)

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-337 p. 179, André Desvigne.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A020884, A020885, A020886, A024354, A024410.

Triangles with 2/a = 1/b + 1/c: A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

for a from 4 to 325 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-bBernard Schott, May 05 2021

Extended and corrected by David W. Wilson

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

Original entry on oeis.org

6, 15, 20, 28, 35, 42, 45, 63, 66, 72, 77, 88, 91, 99, 104, 110, 117, 120, 130, 143, 153, 156, 165, 170, 187, 190, 195, 204, 209, 210, 221, 228, 231, 238, 247, 255, 266, 272, 273, 276, 285, 299, 304, 322, 323, 325, 336, 342, 345, 350, 357, 368, 378, 391, 399

Offset: 1

Clark Kimberling

nonn

k is in this sequence iff A078926(k) > 0.
Also, ordered sides c of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c (A343893). - Bernard Schott, May 06 2021
a(n) are the ordered radii of inscribed circles in squares, from which the tangents to the circles are cut off by primitive Pythagorean triangles. - Alexander M. Domashenko, Oct 17 2024

Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)

Subsequence of A005279.
Cf. A024364, A078926, A020882, A020884, A020885.
Triangles with 2/a = 1/b + 1/c:  A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

isA020886 := proc(an) local r::integer,s::integer ; for r from floor((an/2)^(1/2)) to floor(an^(1/2)) do for s from r-1 to 1 by -2 do if r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA020886(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[ n/2^IntegerExponent[n, 2]]}];
Select[Range[1000], A078926[#]>0&] (* Jean-François Alcover, Mar 23 2020 *)

is(n,f=factor(n))=my(P=apply(i->f[i,1]^f[i,2],[2-n%2..#f~]),nn=2*n); forvec(v=vector(#P,i,[0,1]), my(d=prod(i=1,#v,P[i]^v[i]),d2=d^2); if(d2n, return(1))); 0
list(lim)=my(v=List()); forfactored(n=6,lim\1, if(is(n[1],n[2]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Feb 03 2023

a(n) = A024364(n)/2.

A343891 List of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

Original entry on oeis.org

4, 3, 6, 12, 10, 15, 15, 12, 20, 21, 15, 35, 24, 21, 28, 35, 30, 42, 40, 36, 45, 45, 35, 63, 55, 40, 88, 56, 44, 77, 60, 55, 66, 63, 56, 72, 72, 52, 117, 77, 63, 99, 80, 65, 104, 84, 78, 91, 91, 70, 130, 99, 90, 110, 105, 77, 165, 112, 105, 120, 117, 99, 143, 120, 85, 204, 132, 102, 187

Offset: 1

Bernard Schott, May 03 2021

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
When sides satisfy 2/a = 1/b + 1/c, or a = 2*b*c/(b+c) then a is always the middle side with b < a < c.
Equivalent relations: the heights and sines satisfy 2*h_a = h_b + h_c and 2/sin(A) = 1/sin(B) + 1/sin(C).
Inequalities between sides: a/2 < b < a < c < b*(1+sqrt(2)).

			(4, 3, 6) is the first triple with 2/4 = 1/3 + 1/6 and 6-4 < 3 < 6+4.
The table begins:
   4,  3,  6;
  12, 10, 15;
  15, 12, 20;
  21, 15, 35;
  24, 21, 28;
  35, 30, 42;
  ...

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-337 p. 179, André Desvigne.

Cf. A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

A343892 Side b of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

Original entry on oeis.org

3, 10, 12, 15, 21, 30, 36, 35, 40, 44, 55, 56, 52, 63, 65, 78, 70, 90, 77, 105, 99, 85, 102, 119, 132, 136, 117, 114, 143, 133, 126, 152, 171, 154, 182, 168, 165, 210, 195, 161, 176, 184, 208, 207, 187, 240, 230, 253, 200, 221, 198, 255, 225, 234, 216, 275, 300, 306, 247, 270

Offset: 1

Bernard Schott, May 06 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
The sequence is not increasing because a(7) = 36 > a(8) = 35, but, these sides b are listed in increasing order in A020890.
The first term appearing twice is 330 and corresponds to triples (435, 330, 638) and (460, 330, 759), the second one is 462 and corresponds to triples (483, 462, 506) and (532, 462, 627).
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(4) = 15, because the fourth triple is (21, 15, 35) with side b = 15, satisfying 1/15 = 2/21 - 1/35 and 31-15 < 21 < 31+15.

Cf. A343891 (triples), A020883 (side a), A343893 (side c), A343894 (perimeter).

Cf. A020890 (sides b ordered).

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

a(n) = A343891(n, 2).

A343893 Side c of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

Original entry on oeis.org

6, 15, 20, 35, 28, 42, 45, 63, 88, 77, 66, 72, 117, 99, 104, 91, 130, 110, 165, 120, 143, 204, 187, 170, 156, 153, 221, 247, 195, 228, 266, 209, 190, 238, 210, 273, 285, 231, 255, 368, 336, 345, 304, 322, 391, 272, 299, 276, 425, 357, 450, 323, 400, 414, 513, 350, 325, 342, 475, 459

Offset: 1

Bernard Schott, May 06 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
The sequence is not increasing because a(4) = 35 > a(5) =  28, but, these sides c are listed in increasing order in A020886.
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(3) = 20, because the third triple is (15, 12, 20) with side c = 20, satisfying 1/20 = 2/15 - 1/12 and 15-12 < 20 < 15+12.

Cf. A343891 (triples), A020883 (side a), A343892 (side b), A343894 (perimeter).

Cf. A020886 (sides c ordered).

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

a(n) = A343891(n, 3).

A020890 Ordered set of (b + c - a)/2 as (a,b,c) runs through all primitive Pythagorean triples with a < b < c.

Original entry on oeis.org

3, 10, 12, 15, 21, 30, 35, 36, 40, 44, 52, 55, 56, 63, 65, 70, 77, 78, 85, 90, 99, 102, 105, 114, 117, 119, 126, 132, 133, 136, 143, 152, 154, 161, 165, 168, 171, 176, 182, 184, 187, 195, 198, 200, 207, 208, 210, 216, 221, 225, 230, 234, 240, 247, 253, 255, 260, 261, 270

Offset: 1

Clark Kimberling

nonn

From Bernard Schott, May 06 2021: (Start)
Also, ordered sides b of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c (A343892).
The first term appearing twice is 330 = a(71) = a(72). (End)

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A020889.

Cf. Triangles with 2/a = 1/b + 1/c: A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

a(n) = A020889(n)/2.

Offset corrected to 1 by Ray Chandler, Jan 23 2020

A343895 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the two other sides b and c, i.e., 2/a = 1/b + 1/c with b < a.

Original entry on oeis.org

13, 37, 47, 71, 73, 107, 121, 143, 177, 181, 183, 191, 239, 241, 249, 253, 291, 299, 337, 347, 359, 409, 421, 429, 431, 433, 491, 517, 503, 529, 537, 541, 563, 579, 587, 649, 659, 661, 671, 743, 753, 759, 767, 769, 781, 789, 793, 831, 851, 897, 863, 913, 923, 933, 937, 947, 971, 1033

Offset: 1

Bernard Schott, May 18 2021

nonn

This sequence is the list of ordered terms of A343894, which is not monotonic.
It first differs from A343894 at index 9 where a(9) = 177 while A343894(9) = 183.
Like A343894, all terms are odd.
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(2) = 12 + 10 + 15 = 37, because the second largest perimeter corresponds to triple (12, 10, 15) with relations 2/12 = 1/10 + 1/15 and 15 - 10 < 12 < 15 + 10.

Cf. A343894.
Cf. A343891, A343892, A343893.
Cf. A020883 (sides a), A020886 (sides c), A020890 (sides b).

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A020883 Ordered long legs of primitive Pythagorean triangles.

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A020886 Ordered semiperimeters of primitive Pythagorean triangles.

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A343891 List of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

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A343892 Side b of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

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A343893 Side c of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

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A020890 Ordered set of (b + c - a)/2 as (a,b,c) runs through all primitive Pythagorean triples with a < b < c.

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A343895 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the two other sides b and c, i.e., 2/a = 1/b + 1/c with b < a.

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