A010814 -id:A010814 - cp's OEIS frontend

A334801 Perimeters of Pythagorean triangles whose long leg divides its area.

24, 40, 48, 60, 70, 72, 80, 84, 96, 112, 120, 126, 140, 144, 160, 168, 176, 180, 192, 198, 200, 210, 216, 220, 224, 240, 252, 260, 264, 280, 286, 288, 300, 308, 312, 320, 330, 336, 350, 352, 360, 364, 378, 384, 390, 396, 400, 408, 416, 420, 432, 440, 442, 448, 456, 468

Offset: 1

Wesley Ivan Hurt, May 12 2020

nonn

			a(1) = 24; There is one Pythagorean triangle with perimeter 24, [6,8,10] whose area is 24 and 8|24.

Wikipedia, Integer Triangle
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A010814, A334761, A334799.

A343207 Numbers k such that there exists a unique partition of k into positive integers x,y,z such that x+y, y+z and x+z divide xy, yz and x*z, respectively.

Original entry on oeis.org

6, 12, 15, 18, 20, 28, 35, 36, 40, 54, 56, 63, 70, 75, 77, 78, 88, 91, 99, 100, 102, 104, 108, 114, 117, 130, 138, 143, 153, 154, 162, 170, 174, 175, 176, 182, 184, 186, 187, 189, 190, 196, 200, 208, 209, 221, 222, 238, 245, 246, 247, 258, 261, 266, 272, 282, 286, 297

Offset: 1

Jean-François Alcover, Apr 08 2021

nonn

A subsequence of A005279, except for terms such as 184, 261, 568, 826, 848, ..., which partition into distinct parts.

A variant of A343126.

			15 = 3+6+6 with 3+6 = 9 | 18 and 6+6 = 12 | 36.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A005279, A010814, A343126.

sel[k_] := Select[IntegerPartitions[k, {3}], ({x, y, z} = Sort[#]; Divisible[x y, x+y] && Divisible[y z, y+z] && Divisible[x z, x+z])&];
Reap[For[k = 3, k <= 500, k++, sk = sel[k]; If[Length[sk] == 1, Print[k, " ", Sort[sk[[1]]]]; Sow[k]]]][[2, 1]]

A379600 a(n) is the semiperimeter of the primitive Pythagorean triangle (x(n), y(n), z(n)) with x(n) < y(n) < z(n) and x(n) > x(n-1), y(n) > y(n-1), z(n) > z(n-1), which has the smallest perimeter (if there are several triangles with smallest perimeter: the one of these with the smallest area), starting from a(1) = (3 + 4 + 5)/2 = 6.

Original entry on oeis.org

6, 15, 20, 35, 63, 77, 99, 104, 130, 165, 204, 247, 266, 336, 345, 391, 425, 450, 513, 580, 609, 651, 713, 805, 825, 888, 945, 1036, 1107, 1204, 1271, 1376, 1457, 1530, 1617, 1645, 1764, 1887, 1961, 2014, 2090, 2280, 2337, 2419, 2537, 2562, 2684, 2772, 2990, 3149

Offset: 1

Felix Huber, Feb 15 2025

nonn

Conjecture: There is no primitive Pythagorean triangle that has a smaller semiperimeter than a(n) that can be drawn around the primitive Pythagorean triangle (x(n-1), y(n-1), z(n-1)) without touching it.

Subsequence of A020886.

			(8, 15, 17) is the primitive Pythagorean triangle with semiperimeter a(3) = 20. (20, 21, 29) is the primitive Pythagorean triangle with semiperimeter a(4) = 35 because 20 > 8, 21 > 15, 29 > 17 and there is no other primitive Pythagorean triangle with perimeter <= 70 satisfying this criterium. For example, the primitive Pythagorean triangle (7, 24, 25) has the perimeter 56 but 7 < 8.

Felix Huber, Table of n, a(n) for n = 1..3257
Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A010814, A020886, A118858.

A379600:=proc(S) # to get all terms <= S
    local p,q,i,L,M;
    L:=[];
    M:=[[3,4,5,6,6]];
    for p from 3 to floor((sqrt(4*S+1)-1)/2) do
        for q to min(p-1,S/p-p) do
            if gcd(p,q)=1 and is(p-q,odd) then
                L:=[op(L),[min(p^2-q^2,2*p*q),max(p^2-q^2,2*p*q),p^2+q^2,p*(p+q),(p^2-q^2)*p*q]];
            fi
        od
    od;
    L:=sort(sort(L,(x,y)->x[5]<=y[5]),(x,y)->x[4]<=y[4]);
    for i in L do
        if i[1]>M[nops(M),1] and i[2]>M[nops(M),2] and i[3]>M[nops(M),3] then
            M:=[op(M),i]
        fi
    od;
    return seq(M[i,4],i=1..nops(M))
end proc;
A379600(3149);
# 3 lines above: change 4 to 3 for hypotenuses, to 2 for long legs and to 1 for short legs, to 5 for areas

A334759 Perimeters of Pythagorean triangles with even side lengths.

Original entry on oeis.org

24, 48, 60, 72, 80, 96, 112, 120, 140, 144, 160, 168, 180, 192, 216, 224, 240, 252, 264, 280, 288, 300, 308, 312, 320, 336, 352, 360, 364, 384, 396, 400, 408, 416, 420, 432, 440, 448, 456, 468, 480, 504, 520, 528, 540, 552, 560, 572, 576, 600, 612, 616, 624, 640, 648, 660

Offset: 1

Wesley Ivan Hurt, May 10 2020

nonn

			a(1) = 24; There is one integer-sided right triangle with perimeter 24, [6,8,10] with all side lengths even.

Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A005044, A010814, A334607.

A334760 Perimeters of Pythagorean triangles whose shortest side length divides its perimeter.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 56, 60, 72, 80, 84, 90, 96, 108, 112, 120, 132, 144, 150, 156, 160, 168, 180, 182, 192, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 276, 280, 288, 300, 306, 312, 320, 324, 330, 336, 348, 360, 364, 372, 380, 384, 390, 392, 396, 400

Offset: 1

Wesley Ivan Hurt, May 10 2020

nonn

			a(1) = 12; There is one integer-sided right triangle with perimeter 12, [3,4,5]. Since 3|12, 12 is in the sequence.
a(2) = 24; There is one integer-sided right triangle with perimeter 24, [6,8,10]. Since 6|24, 24 is in the sequence.

Wikipedia, Integer Triangle
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A005044, A010814.

A334790 Perimeters of Pythagorean triangles whose perimeter divides their area.

Original entry on oeis.org

24, 30, 48, 60, 70, 72, 80, 90, 96, 112, 120, 126, 140, 144, 150, 154, 160, 168, 180, 182, 192, 198, 210, 216, 224, 234, 240, 252, 264, 270, 280, 286, 288, 300, 306, 308, 312, 320, 330, 336, 350, 352, 360, 364, 374, 378, 384, 390, 396, 400, 408, 416, 418, 420, 432

Offset: 1

Wesley Ivan Hurt, May 10 2020

nonn

			a(1) = 24; There is one Pythagorean triangle, [6,8,10], with perimeter 24 and area 24. Since 24|24, 24 is in the sequence.

Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A010814.

A348577 Positive integers that are not the perimeter of any integer-sided right triangle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85

Offset: 1

Wesley Ivan Hurt, Oct 23 2021

nonn

Complement of A010814.

Cf. A010814.

seq[max_] := Module[{s = {}, a, b, c}, Do[If[IntegerQ[c = Sqrt[a^2 + b^2]], AppendTo[s, a + b + c]], {a, max}, {b, Floor@Sqrt[a], a}]; Complement[Range[max], s]]; seq[100] (* Amiram Eldar, Oct 24 2021 *)

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A334801 Perimeters of Pythagorean triangles whose long leg divides its area.

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A343207 Numbers k such that there exists a unique partition of k into positive integers x,y,z such that x+y, y+z and x+z divide x*y, y*z and x*z, respectively.

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A334759 Perimeters of Pythagorean triangles with even side lengths.

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A334760 Perimeters of Pythagorean triangles whose shortest side length divides its perimeter.

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A334790 Perimeters of Pythagorean triangles whose perimeter divides their area.

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A348577 Positive integers that are not the perimeter of any integer-sided right triangle.

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Mathematica

A343207 Numbers k such that there exists a unique partition of k into positive integers x,y,z such that x+y, y+z and x+z divide xy, yz and x*z, respectively.