A024364 -id:A024364 - cp's OEIS frontend

A335035 Ordered perimeters of primitive integer triangles with two perpendicular medians.

54, 70, 104, 154, 170, 216, 252, 266, 352, 368, 418, 442, 464, 594, 598, 620, 638, 720, 740, 748, 792, 810, 902, 952, 962, 988, 1054, 1102, 1118, 1134, 1148, 1170, 1216, 1274, 1316, 1376, 1426, 1484, 1512, 1564, 1568, 1598, 1600, 1638, 1702, 1710, 1802, 1836, 1862

Offset: 1

Bernard Schott, May 27 2020

nonn

The study of these integer triangles that have two perpendicular medians was proposed in the problem of Concours Général in 2007 in France (see link).
If medians drawn from A and B are perpendicular in centroid G, then a^2 + b^2 = 5 * c^2 (see Maths Challenge picture in link).
All terms are even because each triple is composed of one even side and two odd sides.
For the corresponding primitive triples and miscellaneous properties, see A335034.

			a(1) = 13 + 19 + 22 = 54 with 19^2 + 22^2 = 5 * 13^2 = 845.

Annales Concours Général, Sujet Concours Général 2007
Maths Challenge, Perpendicular medians, Problem with picture.

Cf. A024364 (perimeters of primitive Pythagorean triangles).

Cf. A335034 (corresponding primitive triples), A335036 (smallest side), A335347 (middle side), A335348 (largest side), A335273 (even side).

lista(nn) = {my(vm = List(), vt); for (u=1, nn, for (v=1, nn, if (gcd(u, v) == 1, vt = 0; if ((u/v > 3) && ((u-3*v) % 5), vt = [2*(u^2-u*v-v^2), u^2+4*u*v-v^2, u^2+v^2]); if ((u/v > 1) && (u/v < 2) && ((u-2*v) % 5), vt = [2*(u^2+u*v-v^2), -u^2+4*u*v+v^2, u^2+v^2]); if ((gcd(vt) == 1), listput(vm, vecsum(vt)));););); vecsort(vm);} \\ Michel Marcus, May 27 2020

a(n) = A335036(n) + A335347(n) + A335348(n).

A024408 Perimeters of more than one primitive Pythagorean triangle.

Original entry on oeis.org

1716, 2652, 3876, 3960, 4290, 5244, 5700, 5720, 6900, 6930, 8004, 8700, 9300, 9690, 10010, 10788, 11088, 12180, 12876, 12920, 13020, 13764, 14280, 15252, 15470, 15540, 15960, 16380, 17220, 17480, 18018, 18060, 18088, 18204, 19092, 19320, 20592, 20868

Offset: 1

David W. Wilson

nonn

a(23) = 14280 is the first perimeter of 3 primitive Pythagorean triangles: {119, 7080, 7081}, {168, 7055, 7057} and {3255, 5032, 5993}. - Jean-François Alcover, Mar 14 2012

			a(1) = 1716 with precisely two primitive Pythagorean triangles (with increasing entries): {195, 748, 773} and {364, 627, 725}. From Ron Knott's link. This is the first example of the family of perimeters 12*b(k)*(b(k) + 2) with b(k) = A007528(k), for k >= 2. See the Bernstein link, p. 234, Theorem 5. a). - _Wolfdieter Lang_, Sep 24 2019

David A. Corneth, Table of n, a(n) for n = 1..10000
Leon Bernstein, Primitive Pythagorean Triples, The Fibonacci Quarterly 20.3 (1982) 227-241.
Ron Knott, Pythagorean Triples and Online Calculators
Lindsay Witcosky, Perimeters of primitive Pythagorean triangles

Cf. A007528, A024364, A103606.

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

1, 5, 140, 385, 2870, 8555, 29370, 42925, 93665, 116795, 149226, 155155, 348551, 380380, 414090, 513590, 1229305, 1801800, 2567895, 2767905, 3873301, 3924830, 5053620, 6970150, 17090486, 18362930, 23396450, 31919165, 39336465, 41791750

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5, ar=3*4/2=6, s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;ar=a*b/2;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,ar/6]],{n,8!}];lst

A381008 Ordered perimeters of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

56, 800, 12416, 197120, 3147776, 50339840, 805339136, 12885032960, 206158954496, 3298536980480, 52776566521856, 844424963686400, 13510799016329216, 216172782650654720, 3458764515968024576, 55340232229718589440, 885443715572418215936, 14167099448746374594560

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A024364.

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A024364.

Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381009 (areas).

[2^(4*n+1) + 2^(2*n+1) + 2^(4*n): n in [1..20]];

A381008[n_] := #*(3*# + 2) & [4^n]; Array[A381008, 20] (* or *)
LinearRecurrence[{20, -64}, {56, 800}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI
```
a(n) = 2^(4*n+1) + 2^(2*n+1) + 2^(4*n)
```

def A381008(n): return (m:=1<<(n<<1))*(2+3*m) # Chai Wah Wu, Feb 13 2025

a(n) = A381005(n) + A381006(n) + A381007(n).
a(n) = 2^(4n+1) + 2^(2n+1) + 2^(4n).
G.f.: 8*(7 - 40*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

Original entry on oeis.org

1, 7916, 35882, 37816, 47491, 128429, 131830, 146471, 154799, 157579, 170219, 174964, 187544, 207829, 208039, 222887, 223142, 262502, 291544, 319825, 327602, 331627, 353857, 476681, 477659, 494207, 522025, 537454, 540682, 558161, 571670

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes,pr=3*4*5=60-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;pr=a*b*c;If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[pr-1]&&PrimeQ[pr+1],AppendTo[lst,n]],{n,3*9!}];lst

A225951 Triangle for perimeters of primitive Pythagorean triangles.

Original entry on oeis.org

12, 0, 30, 40, 0, 56, 0, 70, 0, 90, 84, 0, 0, 0, 132, 0, 126, 0, 154, 0, 182, 144, 0, 176, 0, 208, 0, 240, 0, 198, 0, 234, 0, 0, 0, 306, 220, 0, 260, 0, 0, 0, 340, 0, 380, 0, 286, 0, 330, 0, 374, 0, 418, 0, 462, 312, 0, 0, 0, 408, 0, 456, 0, 0, 0, 552, 0, 390, 0, 442, 0, 494, 0, 546, 0, 598, 0, 650, 420, 0, 476, 0, 532, 0, 0, 0, 644, 0, 700, 0, 756

Offset: 2

Wolfdieter Lang, May 21 2013

See the Hardy-Wright (Theorem 225, p. 190) and Niven-Zuckerman-Montgomery (Theorem 5.5, p. 232) references for primitive Pythagorean triangles.
Here a(n,m) = 0 for non-primitive Pythagorean triangles.
There is a one-to-one correspondence between the values n and m of this number triangle for which a(n,m) does not vanish and primitive solutions of x^2 + y^2 = z^2 with y even, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2. The mirror triangles with x even are not considered here. Therefore a(n,m) = (n^2 - m^2) + 2*n*m + (n^2 + m^2) = 2*n*(n+m) (for these solutions).
The number of non-vanishing entries in row n is A055034(n).
The sequence of the diagonal entries is 2*n*(2*n-1) = 2*A000384(n), n >= 2.
The ordered nonzero entries of this triangle gives A024364.
  Note that all perimeters <= N will certainly be found if one consider all rows n = 2, 3, ..., floor((-1 + sqrt(2*N + 1))/2).
See also A070109(n) for the number of primitive Pythagorean triangles with perimeter n and leg y even.

			The triangle a(n,m) begins:
n\m   1    2   3    4    5    6    7    8    9   10   11
2:   12
3:    0   30
4:   40    0  56
5:    0   70   0   90
6:   84    0   0    0  132
7:    0  126   0  154    0  182
8:  144    0 176    0  208    0  240
9:    0  198   0  234    0    0    0  306
10: 220    0 260    0    0    0  340    0  380
11:   0  286   0  330    0  374    0  418    0  462
12: 312    0   0    0  408    0  456    0    0    0  552
...
The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), therefore, a(2,1) = 3 + 4 + 5 = 12.
The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), therefore, a(7,4) = 33 + 56 + 65 = 154.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.

Cf. A024364 (nonzero, ordered), A225949 (leg sums), A222946 (hypotenuses), A000384 (half of the main diagonal), A070109.

a(n,m) = 2*n*(n+m) if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^{n+m} = -1), otherwise a(n,m) = 0.

A155180 Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

Original entry on oeis.org

3, 15833, 71765, 75633, 94983, 256859, 263661, 292943, 309599, 315159, 340439, 349929, 375089, 415659, 416079, 445775, 446285, 525005, 583089, 639651, 655205, 663255, 707715, 953363, 955319, 988415, 1044051, 1074909, 1081365, 1116323

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes,pr=3*4*5=60-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;pr=a*b*c;If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[pr-1]&&PrimeQ[pr+1],AppendTo[lst,a]],{n,3*9!}];lst

A350038 Numbers that are the perimeter of a primitive 60-degree integer triangle.

Original entry on oeis.org

18, 20, 35, 36, 45, 56, 77, 84, 90, 104, 110, 120, 126, 135, 143, 170, 176, 182, 189, 198, 209, 210, 216, 221, 252, 260, 264, 266, 270, 272, 273, 297, 299, 323, 350, 351, 360, 368, 374, 378, 380, 390, 396, 425, 432, 437, 459, 462, 464, 468, 476, 494, 495, 506, 527, 551, 561, 570, 575, 585, 594, 608, 612

Offset: 1

Seiichi Manyama, Dec 10 2021

nonn

			b(n) = Sum_{k=1..3} A264826(3*n+k-3).
c(n) = Sum_{k=1..3} A201223(3*n+k-3).
b(1) = c(1) = 3+7+8 = 18 = a(1).
b(2) = c(2) = 5+7+8 = 20 = a(2).
b(3) = c(5) = 5+19+21 = 45 = a(5).
b(4) = c(3) = 7+13+15 = 35 = a(3).
b(5) = c(9) = 7+37+40 = 84 = a(8).
b(6) = c(4) = 8+13+15 = 36 = a(4).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A024364, A201223, A264826, A350013, A350039.

def A(n)
  ary = []
  (1..n).each{|i|
    (i + 1..n).each{|j|
      if i.gcd(j) == 1 && (i - j) % 3 > 0
        x, y, z = j * j, i * j, i * i
        ary << 2 * x + 5 * y + 2 * z
        ary << 3 * x + 3 * y
      end
    }
  }
  ary
end
p A(20).uniq.sort[0..100]

A093507 Perimeter of primitive Pythagorean triangles sorted on hypotenuse A020882(n).

Original entry on oeis.org

12, 30, 40, 56, 70, 84, 90, 126, 132, 144, 154, 176, 198, 182, 208, 234, 220, 260, 240, 286, 330, 312, 306, 340, 374, 408, 390, 380, 442, 418, 456, 420, 476, 494, 532, 462, 510, 546, 570, 544, 608, 552, 598, 644, 672, 690, 646, 714, 736, 650, 700, 684, 782

Offset: 1

Lekraj Beedassy, May 14 2004

nonn

For ordered perimeter of primitive Pythagorean triangle see A024364.

Michael Somos, Table of primitive Pythagorean triples and related parameters

a(n)=2*A093508.

A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.

Original entry on oeis.org

18, 48, 60, 90, 100, 140, 144, 180, 210, 270, 280, 288, 294, 320, 360, 378, 448, 462, 480, 594, 600, 648, 660, 720, 728, 756, 858, 900, 900, 924, 980, 1008, 1008, 1078, 1080, 1120, 1170, 1210, 1260, 1344, 1496, 1530, 1530, 1568, 1584, 1584, 1680, 1700, 1728

Offset: 1

Alexandre Wajnberg, May 02 2005

			a(28) = 900 = (18+80+82) + (18*80/2) for 18*18 + 80*80 = 82*82.
a(29) = 900 = (25+60+65) + (25*60/2) for 25*25 + 60*60 = 65*65.
a(32) = 1008 = (24+70+74) + (24*70/2) for 24*24 + 70*70 = 74*74.
a(33) = 1008 = (36+48+60) + (36*48/2) for 36*36 + 48*48 = 60*60.

Giovanni Resta, Table of n, a(n) for n = 1..10711 (terms < 10^7)
Douglas Butler, Pythagorean triples [including multiples] up to 2100, Oundle School iCT Training Centre.
Ron Knott, Pythagorean triples.

Cf. A103605, A103606, A024364, A024406, A105521.

L = {}; mx = 1728; Do[ Do[ If[ IntegerQ[z = Sqrt[x^2 + y^2]], v = x y/2 + x + y + z; If[v <= mx, AppendTo[L, v], Break[]]], {y, x-1}], {x, 4, 4 + (2 mx^2)^(1/3)}]; Sort@ L (* Giovanni Resta, Mar 16 2020 *)

T. = 0                        ;  S = ''
do C = 1 to 999               ;  H = C*C
   do D = 1 to C              ;  I = D*D
      do E = 1 to D           ;  J = E*E
         if I + J < H   then  iterate E
         if I + J = H   then  do
            K = T.0 + 1       ;  T.0 = K
            P = C + D + E     ;  A = ( D * E ) / 2
            T.K = right( A + P, 6 )
            T.K = T.K '=' A '+' P '(' E '+' D '+' C ')'
         end
         leave E
      end E
   end D
end C
call KWIK 'T.' /* sort by A+P for area A and perimeter P */
Y = 0
do N = 1 to T.0 while length( S ) < 255
   X = word( T.N, 1 )         ;  say T.N
   if X <= Y   then  say 'dupe:' N - 1 N ':' Y X
   S = S || ', ' || X         ;  Y = X
end N
say substr( S, 3 )            /* Frank Ellermann, Mar 02 2020 */

Corrected and extended by Frank Ellermann, Mar 02 2020

cp's OEIS Frontend

A335035 Ordered perimeters of primitive integer triangles with two perpendicular medians.

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A024408 Perimeters of more than one primitive Pythagorean triangle.

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A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes.

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A381008 Ordered perimeters of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes, pr=a*b*c, pr-+1 are primes.

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A225951 Triangle for perimeters of primitive Pythagorean triangles.

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A155180 Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes, pr=a*b*c, pr-+1 are primes.

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A350038 Numbers that are the perimeter of a primitive 60-degree integer triangle.

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A093507 Perimeter of primitive Pythagorean triangles sorted on hypotenuse A020882(n).

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A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

A155180 Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.