A046084 -id:A046084 - cp's OEIS frontend

A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

12, 20, 24, 28, 36, 40, 45, 44, 48, 52, 60, 56, 60, 72, 72, 68, 75, 63, 84, 76, 80, 90, 84, 88, 105, 92, 105, 96, 120, 120, 104, 120, 108, 112, 132, 105, 116, 120, 144, 124, 144, 135, 132, 156, 136, 150, 126, 140, 168, 168, 180, 148, 175, 165, 152, 156, 168, 180

Offset: 1

Felix Huber, Aug 19 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

			The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 12 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A046084.

Cf. A366675, A380073, A386307, A386309, A386944.

A386308:=proc(N) # To get all terms with hypotenuses <= N
    local i,l,m,u,v,r,x,y,z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            x:=min(2*u*v,u^2-v^2);
            y:=max(2*u*v,u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z,y,x]};
            if gcd(u,v)=1 and is(u-v,odd) then
                l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i,2],i=1..nops(r));
end proc;
A386308(1000);

a(n) = sqrt(A386307(n)^2 - A386309(n)^2).

{A046084(n)} = {a(n)} union {A046087(n)} union {A386944(n)}.

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), ordered by increasing hypotenuse (A386943).

Original entry on oeis.org

8, 16, 24, 30, 32, 36, 48, 48, 42, 60, 70, 64, 80, 72, 96, 96, 90, 84, 108, 120, 100, 112, 126, 120, 110, 140, 135, 128, 160, 154, 168, 160, 144, 144, 192, 198, 192, 180, 182, 216, 224, 168, 216, 240, 196, 200, 234, 224, 252, 189, 240, 210, 286, 288, 220, 280, 280

Offset: 1

Felix Huber, Aug 24 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 8 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A046084.

Cf. A046087, A366675, A380073, A386308, A386943, A386945.

A386944:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,2],i=1..nops(l));
end proc;
A386944(296);

a(n) = sqrt(A386943(n)^2 - A386945(n)^2).

{A046084(n)} = {a(n)} union {A046087(n)} union {A386308(n)}.

A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 0, 5, 0, 9, 0, 8, 0, 0, 12, 0, 0, 0, 0, 7, 10, 0, 0, 20, 18, 0, 0, 0, 16, 21, 0, 12, 0, 15, 24, 9, 0, 0, 0, 27, 0, 0, 0, 0, 14, 24, 20, 28, 0, 33, 0, 0, 40, 0, 36, 11, 0, 0, 0, 16, 0, 0, 32, 0, 42, 0, 0, 48, 24, 21, 0, 0, 30, 0, 48, 0, 18, 0, 0, 13, 0, 60, 0, 39, 54

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) = 3 as the right triangle with sides (3, 4, 5) has hypotenuse n = 5 smallest side a(5) = 3. This is the smallest side a right triangle with integer sides and hypotenuse 5 can have. - _David A. Corneth_, Apr 10 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108708.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_]:=Block[{k=n-1,m=Sqrt[n/2],a},While[k>m&&!IntegerQ[(a=Sqrt[n^2-k^2])],k--];If[k<=m,0,a]];Table[f[n],{n,90}]

first(n) = {my(lh = List(), res = vector(n, i, oo)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = vecmin([res[i*(u^2+v^2)], i*(u^2 - v^2), i*2*u*v]))))); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

Extended by Ray Chandler, Dec 20 2011

A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 12, 0, 12, 0, 15, 0, 0, 16, 0, 0, 0, 0, 24, 24, 0, 0, 21, 24, 0, 0, 0, 30, 28, 0, 35, 0, 36, 32, 40, 0, 0, 0, 36, 0, 0, 0, 0, 48, 45, 48, 45, 0, 44, 0, 0, 42, 0, 48, 60, 0, 0, 0, 63, 0, 0, 60, 0, 56, 0, 0, 55, 70, 72, 0, 0, 72, 0, 64, 0, 80, 0, 0, 84, 0, 63, 0

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) is 4 as the maximum side (other than the hypotenuse) a right triangle with integer sides and hypotenuse 5 can have.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108707.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_] := Block[{k = n - 1, m = Sqrt[n/2]}, While[k > m && !IntegerQ[Sqrt[n^2 - k^2]], k-- ]; If[k <= m, 0, k]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Jun 21 2005 *)

first(n) = {my(lh = List(), res = vector(n)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = max(res[i*(u^2+v^2)], max(i*(u^2 - v^2), i*2*u*v)); ); ); ); ); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

More terms from Robert G. Wilson v, Jun 21 2005

A088896 Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.

Original entry on oeis.org

125, 1000, 2197, 3375, 4913, 8000, 15625, 17576, 24389, 27000, 39304, 42875, 50653, 59319, 64000, 68921, 91125, 125000, 132651, 140608, 148877, 166375, 195112, 216000, 226981, 274625, 314432, 343000, 389017, 405224, 421875, 474552, 512000

Offset: 1

Lekraj Beedassy, Nov 28 2003

nonn

The set of values for the integral-widths corridors and longest ladder are merely the cubes of Pythagorean triples, viz. (A046083, A046084, A009000).
The corridors' widths may be parametrically expressed as d*(sin x)^3 and d*(cos x)^3, for a longest ladder length d making an angle x with one of the corridors.
A given ladder, however, is maximum-corner-bending for a family of infinite pairs of perpendicular corridor widths and that the envelope of the maximum bending positions is that of a sliding rod against the outer wall, which is a branch of an astroid or four-cusped hypocycloid.

E. Mendelson, 3000 Solved Problems in Calculus, Chapter 16 Problem 16.56 pp. 131, Mc Graw-Hill 1988.
M. Spiegel, Theory and Problems of Advanced Calculus, Chapter 4 Problem 40 pp. 75, Mc Graw-Hill 1974.

C. Azeredo, The Ladder Problem
L. Husch and M. Szapiel, The Longest Ladder
M. Kantor, Knox College, Puzzle of the Week
J. J. O'Connor and E. R. Robertson, Astroid
T. Sillke, longest ladder
D. Sjerve, Solution to problem No.3
W. H. Steeb, Solved Problem
Eric Weisstein's World of Mathematics, Astroid

a(n)=d^3, where d=A009003(n).

A096472 Numbers containing squares of Pythagorean triples in their divisor set.

Original entry on oeis.org

3600, 7200, 10800, 14400, 18000, 21600, 25200, 28800, 32400, 36000, 39600, 43200, 46800, 50400, 54000, 57600, 61200, 64800, 68400, 72000, 75600, 79200, 82800, 86400, 90000, 93600, 97200, 100800, 104400, 108000, 111600, 115200, 118800, 122400, 126000, 129600, 133200

Offset: 1

Reinhard Zumkeller, Aug 13 2004

a(n) = m * (A046083(k)*A046084(k)*A009000(k))^2 for appropriate, not necessarily unique m and k.

			5^2 + 12^2 = 13^2: 5^2, 12^2 and 13^2 are divisors of 608400 = (13*5*3*2^2)^2, therefore 608400 is a term.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. Pythagorean triples: A046083, A046084, A009000.

Cf. A044102, A094519, A169823.

Range[50]*3600 (* Paolo Xausa, Jul 01 2025 *)

my(x='x+O('x^38)); Vec(3600*x/(1-x)^2) \\ Elmo R. Oliveira, Jun 30 2025

a(n) = n*60^2.
From Elmo R. Oliveira, Jun 30 2025: (Start)
G.f.: 3600*x/(1-x)^2.
E.g.f.: 3600*x*exp(x).
a(n) = 60*A169823(n) = 100*A044102(n).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Name clarified by Tanya Khovanova, Jul 05 2021

More terms from Elmo R. Oliveira, Jun 30 2025

A342583 Numbers k such that prime(k) is the hypotenuse of a Pythagorean triple where one leg is also prime.

Original entry on oeis.org

3, 6, 18, 42, 82, 271, 284, 369, 445, 682, 1069, 1193, 1900, 2241, 3894, 6137, 7108, 8164, 9658, 10126, 12645, 14842, 14936, 17913, 18420, 19480, 23893, 24605, 28959, 32913, 36279, 40847, 43936, 44559, 45500

Offset: 1

Ivan N. Ianakiev, Mar 16 2021

nonn

In such a triangle, the leg that is not prime is always the largest one and is equal to prime(k)-1; these even legs are in A067755. E.g. for a(2) = 6, prime(6) = 13 and the corresponding Pythagorean triple is (5, 12, 13). - Bernard Schott, Apr 03 2021

			a(1) = 3, since prime(3) = 5 is the hypotenuse of the triple (3,4,5).

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A067756 (the hypotenuses).

Cf. A009000, A046083, A046084, A067755.

R:= NULL: count:= 0:
p:= 2:
while count < 100 do
  p:= nextprime(p); n:= (p-1)/2; q:= 2*n^2+2*n+1;
  if isprime(q) then
    count:= count+1; r:= numtheory:-pi(q); R:= R, r;
  fi
od:
R; # Robert Israel, Mar 22 2021

PrimePi[Take[Cases[Import["https://oeis.org/A067756/b067756.txt","Table"],{,}][[All,2]],100]]

A342858 a(n) is the least integer h such that there exists a Pythagorean triple (x, y, h) that satisfies f(x)+f(y)+f(h)=n where f(m)=A176774(m) is the smallest polygonality of m; a(n) = 0 if no such h exists.

Original entry on oeis.org

13530, 136, 35, 5, 4510, 10, 100, 45, 51, 1404

Offset: 9

Michel Marcus, Mar 26 2021

a(19) > 10^9 if it exists.

It appears that the triples whose sum is 10 (as in the 2nd example below) have legs n^6 = A001014(n), (n^8 - n^4)/2 = A218131(n+1)/2 and (n^8 + n^4)/2 = A071231(n) for n >= 2; they consist of 2 triangular numbers and 1 square number. - Michel Marcus, Apr 12 2021

			a(9)  = 13530 with A176774([8778, 10296, 13530]) = [3,3,3].
a(10) = 136   with A176774([64, 120, 136])       = [4,3,3].
a(11) = 35    with A176774([21, 28, 35])         = [3,3,5].
a(12) = 5     with A176774([3, 4, 5])            = [3,4,5].
a(13) = 4510  with A176774([2926, 3432, 4510])   = [3,5,5].
a(14) = 10    with A176774([6, 8, 10])           = [3,8,3].
a(15) = 100   with A176774([28, 96, 100])        = [3,8,4].
a(16) = 45    with A176774([27, 36, 45])         = [10,3,3].
a(17) = 51    with A176774([45, 24, 51])         = [3,9,5].
a(18) = 1404  with A176774([540, 1296, 1404])    = [7,4,7].

Michel Marcus, Table of n, a(n) for n = 9..10000, with 0 for a(19)

Cf. A009000, A046083, A046084, A176774.
Cf. A213188 (see 2nd comment).
Cf. A245646, A245647, A245648, A342491.
Cf. A001014, A071231, A218131.

tp(n) = if (n<3, [n], my(v=List()); fordiv(2*n, k, if(k<2, next); if(k==n, break); my(s=(2*n/k-4+2*k)/(k-1)); if(denominator(s)==1, listput(v, s))); v = Vec(v); v[#v]); \\ A176774
vsum(v) = vecsum(apply(tp, v));
lista(limp, lim) = {my(vr = vector(limp)); for(u = 2, sqrtint(lim), for(v = 1, u, if (u*u+v*v > lim, break); if ((gcd(u,v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u*u+v*v) > lim, break); my(w = [i*(u*u - v*v), i*2*u*v, i*(u*u+v*v)]); my(h = i*(u*u+v*v)); my(sw = vsum(w)); if (sw <= limp, if (vr[sw] == 0, vr[sw] = h, if (h < vr[sw], vr[sw] = h))););););); vector(#vr - 8, k, vr[k+8]);}
lista(80, 15000) \\ Michel Marcus, Apr 16 2021

A235990 Consider all Pythagorean triples (X,Y,Z) consisting of refactorable numbers (A033950) only, ordered by increasing Z. This sequence gives the values of Z.

Original entry on oeis.org

104, 204, 348, 480, 488, 492, 600, 732, 936, 1248, 1356, 1360, 1440, 1448, 1644, 1788, 2000, 2172, 2196, 2700, 2784, 2824, 3060, 3084, 3228, 3360, 3368, 3372, 3552, 3712, 3744, 3816, 3924, 4080, 4240, 4392, 4500, 4500, 4812, 5052, 5088, 5220, 5280, 5856, 6000

Offset: 1

Ivan N. Ianakiev, Jan 18 2014

nonn

			104^2 = 40^2 + 96^2, i.e. A033950(17)^2 = A033950(9)^2 + A033950(16)^2.

Cf. A033950, A009000, A046083, A046084.

More terms from Michel Marcus, Jan 21 2014

cp's OEIS Frontend

A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

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A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), ordered by increasing hypotenuse (A386943).

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A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

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A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

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A088896 Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.

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A096472 Numbers containing squares of Pythagorean triples in their divisor set.

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A342583 Numbers k such that prime(k) is the hypotenuse of a Pythagorean triple where one leg is also prime.

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A342858 a(n) is the least integer h such that there exists a Pythagorean triple (x, y, h) that satisfies f(x)+f(y)+f(h)=n where f(m)=A176774(m) is the smallest polygonality of m; a(n) = 0 if no such h exists.

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A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), ordered by increasing hypotenuse (A386943).