A088319 -id:A088319 - cp's OEIS frontend

A088515 Values j of pairs (j,k) that generate A088319(n).

1, 1, 5, 1, 7, 7, 1, 1, 9, 9, 3, 1, 11, 11, 3, 11, 1, 5, 13, 13, 1, 13, 3, 1, 15, 15, 5, 3, 15, 1, 5, 17, 17, 7, 17, 17, 1, 5, 7, 3, 19, 19, 19, 19, 5, 1, 19, 3, 21, 21, 7, 1, 21, 21, 7, 5, 23, 9, 23, 1, 23, 23, 3, 23, 7, 5, 9, 1, 23, 3, 25, 25, 7, 25, 25, 5, 25, 1, 7, 9, 27, 25, 27, 27, 5, 11, 1

Offset: 1

Lekraj Beedassy, Nov 14 2003

nonn

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

Cf. A088319, A088516, A088546, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, j, k}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 2]]& (* Jean-François Alcover, Mar 05 2020 *)

Corrected and extended by Ray Chandler, Nov 16 2003

A088516 Values k of pairs (j,k) that generate A088319(n).

Original entry on oeis.org

4, 6, 2, 8, 2, 4, 10, 12, 4, 2, 14, 14, 4, 6, 16, 2, 16, 18, 6, 4, 18, 2, 20, 20, 8, 4, 22, 22, 2, 22, 24, 6, 8, 24, 4, 2, 24, 26, 26, 26, 8, 6, 10, 4, 28, 26, 2, 28, 8, 10, 30, 28, 4, 2, 32, 32, 10, 32, 8, 30, 12, 6, 32, 4, 34, 34, 34, 32, 2, 34, 12, 8, 36, 14, 6, 36, 4, 34, 38, 38, 10, 2

Offset: 1

Lekraj Beedassy, Nov 14 2003

nonn

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

Cf. A088319, A088515, A088546, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, j, k}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 3]]& (* Jean-François Alcover, Mar 05 2020 *)

Corrected and extended by Ray Chandler, Nov 16 2003

A089545 Values e of pairs (e,f) that generate A088319(n).

Original entry on oeis.org

5, 13, 17, 25, 37, 29, 41, 61, 53, 65, 65, 85, 85, 73, 89, 101, 113, 97, 109, 125, 145, 145, 149, 181, 137, 173, 157, 185, 197, 221, 193, 205, 185, 169, 229, 257, 265, 233, 205, 269, 241, 265, 221, 293, 277, 313, 325, 317, 305, 281, 289, 365, 365, 401, 337, 377

Offset: 1

Ray Chandler, Nov 16 2003

nonn

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

Cf. A088319, A088515, A088516, A088546, A089546-A089552, A089554-A089558.

A089546 Values f of pairs (e,f) that generate A088319(n).

Original entry on oeis.org

4, 6, 10, 8, 14, 28, 10, 12, 36, 18, 42, 14, 44, 66, 48, 22, 16, 90, 78, 52, 18, 26, 60, 20, 120, 60, 110, 66, 30, 22, 120, 102, 136, 168, 68, 34, 24, 130, 182, 78, 152, 114, 190, 76, 140, 26, 38, 84, 168, 210, 210, 28, 84, 42, 224, 160, 230, 288, 184, 30, 276, 138, 96, 92

Offset: 1

Ray Chandler, Nov 16 2003

nonn

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

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089558.

A089552 Sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

Original entry on oeis.org

49, 289, 529, 961, 2209, 1681, 2401, 5041, 5329, 6241, 7921, 9409, 12769, 10609, 14161, 14161, 16129, 18769, 22801, 25921, 25921, 27889, 36481, 39601, 37249, 47089, 47089, 54289, 49729, 58081, 69169, 73441, 66049, 57121, 78961, 82369

Offset: 1

Ray Chandler, Nov 16 2003

nonn

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

Cf. A088319, A088515, A088516, A088546, A089545-A089551, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, (j^2 - k^2/2)^2}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 2]]& (* Jean-François Alcover, Mar 05 2020 *)

A088546 Square root of sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

Original entry on oeis.org

7, 17, 23, 31, 47, 41, 49, 71, 73, 79, 89, 97, 113, 103, 119, 119, 127, 137, 151, 161, 161, 167, 191, 199, 193, 217, 217, 233, 223, 241, 263, 271, 257, 239, 281, 287, 287, 313, 289, 329, 329, 343, 311, 353, 367, 337, 359, 383, 409, 391, 401, 391, 433, 439, 463

Offset: 1

Lekraj Beedassy, Nov 17 2003

nonn

Numbers whose square is the sum of the legs of primitive Pythagorean triangles with hypotenuse A088319(n).

			31 is in the sequence because it is associated with the primitive Pythagorean triangle (400,561,689) where 400+561=31^2.

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

Cf. A088319, A088515, A088516, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, Abs[j^2 - k^2/2]}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 2]]& (* Jean-François Alcover, Mar 05 2020 *)

a(n) = abs(j^2 - k^2/2), where j=A088515(n), k=A088516(n).

a(n) = sqrt(A089552(n)).

More terms from Ray Chandler, Nov 16 2003

A089554 a(n)=(A088515(n)+1)/2.

Original entry on oeis.org

1, 1, 3, 1, 4, 4, 1, 1, 5, 5, 2, 1, 6, 6, 2, 6, 1, 3, 7, 7, 1, 7, 2, 1, 8, 8, 3, 2, 8, 1, 3, 9, 9, 4, 9, 9, 1, 3, 4, 2, 10, 10, 10, 10, 3, 1, 10, 2, 11, 11, 4, 1, 11, 11, 4, 3, 12, 5, 12, 1, 12, 12, 2, 12, 4, 3, 5, 1, 12, 2, 13, 13, 4, 13, 13, 3, 13, 1, 4, 5, 14, 13, 14, 14, 3, 6, 1, 2, 5, 4, 14, 6

Offset: 1

Ray Chandler, Nov 16 2003

nonn

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

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089555-A089558.

A089558 a(n)=A089551(n)/2.

Original entry on oeis.org

2, 21, 35, 68, 161, 14, 155, 294, 306, 423, 483, 497, 902, 231, 984, 869, 776, 315, 1209, 1898, 1143, 1547, 2670, 1610, 1020, 3390, 2585, 3927, 2505, 2189, 4380, 5253, 3332, 84, 5474, 3791, 2892, 6695, 2093, 7449, 6764, 8607, 2945, 8246, 9590, 3731, 5453

Offset: 1

Ray Chandler, Nov 16 2003

nonn

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

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089557.

A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.

Original entry on oeis.org

35, 41, 140, 164, 205, 221, 315, 369, 389, 391, 560, 656, 689, 775, 820, 875, 884, 1025, 1189, 1260, 1476, 1556, 1564, 1565, 1625, 1715, 1739, 1781, 1845, 1855, 1989, 2009, 2240, 2624, 2756, 2835, 3100, 3280, 3321, 3500, 3501, 3519, 3536, 3865, 3869, 4100, 4105

Offset: 1

Felix Huber, Jan 18 2025

nonn

Corresponding long legs in A380073, short legs in A380074.

Subsequence of A009000 and supersequence of A088319.

			35 is in the sequence because 21^2 + 28^2 = 35^2 and 21 + 28 = 7^2.
206125 is twice in the sequence because 31525^2 + 203700^2 = 206125^2 and 31525 + 203700 = 485^2 as well as 94588^2 + 183141^2 = 206125^2 and 94588 + 183141 = 527^2.

Felix Huber, Table of n, a(n) for n = 1..10001
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A000290, A007913, A009000, A088319, A379830, A380073, A380074.

# Calculates the first 10001 terms
A380072:=proc(M)
    local i,m,p,q,r,w,L;
    L:=[];
    m:=M^2+2*M+2;
    for p from 2 to M do
        for q to p-1 do
            if gcd(p,q)=1 and (is(p,even) or is(q,even)) then
                r:=1;
                for i in ifactors(p^2-q^2+2*p*q)[2] do
                    if is(i[2],odd) then
                        r:=r*i[1]
                    fi
                od;
                w:=r*(p^2+q^2);
                if w<=m then
                    L:=[op(L),seq(i^2*w,i=1..floor(sqrt(m/w)))]
                fi
            fi
        od
    od;
    return op(sort(L))
end proc;
A380072(4330);

A089551 Radius of inscribed circle within primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

Original entry on oeis.org

4, 42, 70, 136, 322, 28, 310, 588, 612, 846, 966, 994, 1804, 462, 1968, 1738, 1552, 630, 2418, 3796, 2286, 3094, 5340, 3220, 2040, 6780, 5170, 7854, 5010, 4378, 8760, 10506, 6664, 168, 10948, 7582, 5784, 13390, 4186, 14898, 13528, 17214, 5890, 16492, 19180

Offset: 1

Ray Chandler, Nov 16 2003

nonn

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

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089558.

cp's OEIS Frontend

A088515 Values j of pairs (j,k) that generate A088319(n).

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A088516 Values k of pairs (j,k) that generate A088319(n).

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A089545 Values e of pairs (e,f) that generate A088319(n).

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A089546 Values f of pairs (e,f) that generate A088319(n).

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A089552 Sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

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Mathematica

A088546 Square root of sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

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A089554 a(n)=(A088515(n)+1)/2.

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A089558 a(n)=A089551(n)/2.

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A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.

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Maple

A089551 Radius of inscribed circle within primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

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