cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A094904 Least hypotenuse of primitive Pythagorean triangles exceeding a leg by A096033(n).

Original entry on oeis.org

5, 5, 13, 17, 25, 37, 41, 65, 61, 85, 101, 113, 145, 145, 181, 197, 221, 257, 265, 313, 325, 365, 401, 421, 485, 481, 545, 577, 613, 677, 685, 761, 785, 841, 901, 925, 1025, 1013, 1105, 1157, 1201, 1297, 1301, 1405, 1445, 1513, 1601, 1625, 1765, 1741, 1861
Offset: 1

Views

Author

Lekraj Beedassy, Jun 16 2004

Keywords

Crossrefs

Cf. A096033.

Extensions

Corrected and extended by Ray Chandler, Jun 18 2004

A321067 Considering Pythagorean triple (a,b,c) with a < b < c, least a such that there exists a primitive triple where c - b is the n-th term of A096033.

Original entry on oeis.org

3, 8, 20, 33, 48, 65, 88, 119, 140, 204, 207, 252, 297, 336, 396, 429, 540, 555, 616, 696, 731, 832, 893, 1036, 1113, 1140, 1248, 1311, 1428, 1525, 1692, 1748, 1809, 1960, 2059, 2184, 2325, 2508, 2576, 2739, 2832
Offset: 1

Views

Author

Jeffrey Burch, Oct 26 2018

Keywords

Examples

			a(2) = 8 because in the primitive triple (8,15,17), c - b = A096033(2) = 2 and no smaller a yields a primitive triple where a < b < c and c - b = 2.

Links

Crossrefs

Cf. A096033, A094904.

Programs

  • Mathematica
    nmax = 100; kmax = 2;
A096033 = Union[2 Range[nmax]^2, (2 Range[0, Ceiling[nmax/Sqrt[2]]]+1)^2];
r[n_, k_] := Module[{a, b, c}, {a, b, c} /. {ToRules[Reduce[0 < a < b < c && c - b == A096033[[n]] && a^2 + b^2 == c^2, {a, b, c}, Integers] /. C[1] -> k]}];
a[n_] := a[n] = SelectFirst[Flatten[Table[r[n, k], {k, 1, kmax}], 1], GCD @@ # == 1 &] // First;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 01 2019 *)

A118962 Difference between short leg and hypotenuse in primitive Pythagorean triangles, sorted on hypotenuse (A020882), then on long leg (A046087).

Original entry on oeis.org

2, 8, 9, 18, 9, 25, 32, 25, 50, 32, 49, 25, 49, 72, 50, 32, 81, 49, 98, 81, 49, 121, 128, 98, 72, 50, 121, 162, 81, 128, 98, 169, 72, 121, 81, 200, 169, 128, 121, 225, 169, 242, 200, 162, 121, 128, 225, 98, 169, 288, 242, 121, 289, 162, 169, 128, 289, 338, 121, 225, 242
Offset: 1

Views

Author

Lekraj Beedassy, May 07 2006

Keywords

Comments

Entries take only values appearing in A096033.

Links

Crossrefs

Cf. A118961, A020882, A046086, A096033, A120682.

Formula

a(n) = A020882(n) - A046086(n) = A118961(n) + A120682(n). - Paul Curtz, Dec 11 2008

Extensions

Corrected and extended by Joshua Zucker, May 11 2006

A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 2, 3, 1, 2, 1, 0, 1, 0, 5, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 1, 2, 1, 0, 2, 4, 7, 0, 1, 0, 4, 0, 2, 0, 1, 0, 1, 0, 1, 2, 5, 0, 1, 0, 1, 0, 1, 0, 8, 0, 1, 3, 1, 0, 1, 0, 2, 6, 1, 0, 1, 0, 1, 0
Offset: 0

Views

Author

Felix Huber, Jan 07 2025

Keywords

Comments

The difference between the hypotenuse and the short leg of a primitive Pythagorean triple (p^2 - q^2, 2*p*q, p^2 + q^2) (where p > q are coprimes and not both odd) is d = max(2*q^2, (p - q)^2). For every of these primitive Pythagorean triples whose d divides n, there is a Pythagorean triple with w - u = n. Therefore d <= n and it follows that 1 <= q <= sqrt(n/2) and q + 1 <= p <= q + sqrt(n), which means that there is a finite number of Pythagorean triples with w - u = n.

Examples

			The a(18) = 4 Pythagorean triples are (27, 36, 45), (16, 30, 34), (40, 42, 58), (7, 24, 25) because 45 - 27 = 34 - 16 = 58 - 40 = 25 - 7 = 18.
See also linked Maple program "Pythagorean triples for which w - u = n".

Links

Crossrefs

Cf. A024361, A024362, A024363, A046079, A046080, A046081, A096033, A224921.

Programs

  • Maple
    A379830:=proc(n)
    local a,p,q;
    a:=0;
    for q to isqrt(floor(n/2)) do
        for p from q+1 to q+isqrt(n) do
            if igcd(p,q)=1 and (is(p,even) or is(q,even)) and n mod max((p-q)^2,2*q^2)=0 then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A379830(n),n=0..87);

A118961 Difference between long leg and hypotenuse in primitive Pythagorean triangles, sorted on hypotenuse (A020882), then on long leg (A046087).

Original entry on oeis.org

1, 1, 2, 1, 8, 2, 1, 8, 1, 9, 2, 18, 8, 1, 9, 25, 2, 18, 1, 8, 32, 2, 1, 9, 25, 49, 8, 1, 32, 9, 25, 2, 49, 18, 50, 1, 8, 25, 32, 2, 18, 1, 9, 25, 50, 49, 8, 81, 32, 1, 9, 72, 2, 49, 50, 81, 8, 1, 98, 32, 25, 49, 72, 2, 18, 1, 121, 9, 25, 49, 8, 98, 32, 81, 121, 1, 9, 2, 25, 18, 128, 49, 50
Offset: 1

Views

Author

Lekraj Beedassy, May 07 2006

Keywords

Comments

Entries take only values appearing in A096033.

Links

Crossrefs

Cf. A118962, A020882, A046087, A096033, A120682.

Formula

a(n) = A020882(n) - A046087(n) = A118962(n) - A120682(n). - Ray Chandler, Nov 24 2019

Extensions

More terms from Joshua Zucker, May 11 2006

A227744 Squares that occur in A173318.

Original entry on oeis.org

0, 1, 4, 9, 144, 169, 256, 289, 324, 361, 400, 576, 625, 784, 841, 900, 1024, 1156, 1225, 1521, 1681, 2116, 2304, 2401, 3721, 4225, 5184, 5329, 6241, 7225, 8281, 8464, 8649, 9604, 10000, 10816, 12100, 18225, 18496, 21904, 24025, 24336, 24649, 26244, 28900, 31329
Offset: 1

Views

Author

Antti Karttunen, Jul 25 2013, proposed by Jonathan Vos Post in Comments section of A173318

Keywords

Links

Crossrefs

Cf. A227745 (gives the square roots of these terms).
All values A096033(n)*(2^(A096033(n)-1)) occur here. - Antti Karttunen, Jul 29 2013

Programs

  • Mathematica
    seq = {0}; n = 0; s = 0; While[Length[seq] < 100,
s += Length[Length /@ Split[IntegerDigits[++n, 2]]]; If[IntegerQ@ Sqrt@ s, AppendTo[seq, s]]]; seq (* Giovanni Resta, Jul 27 2013 *)
  • Scheme
    (define (A227744 n) (A173318 (A227743 n)))

Formula

a(n) = A173318(A227743(n)).

A308222 Numbers that are the perimeter of a primitive Heronian isosceles triangle.

Original entry on oeis.org

16, 18, 36, 50, 64, 98, 100, 144, 162, 196, 242, 256, 324, 338, 400, 450, 484, 576, 578, 676, 722, 784, 882, 900, 1024, 1058, 1156, 1250, 1296, 1444, 1458, 1600, 1682, 1764, 1922, 1936, 2116, 2178, 2304, 2450, 2500, 2704, 2738, 2916, 3042, 3136
Offset: 1

Views

Author

Peter Kagey, May 16 2019

Keywords

Comments

A primitive Heronian triangle is a triangle with integer sides and area, where the side lengths do not share a common divisor.

Examples

			Table illustrating first six terms of the sequence:
  perimeter |   sides    | area
  ----------+------------+-----
      16    |  (5,5,6)   |  12
      18    |  (5,5,8)   |  12
      36    | (10,13,13) |  60
      50    | (13,13,24) |  60
      50    | (16,17,17) | 120
      64    | (14,25,25) | 168
      64    | (17,17,30) | 120
      98    | (24,37,37) | 420
      98    | (25,25,48) | 168
      98    | (29,29,40) | 420

Links

Crossrefs

Cf. A046790, A096033, A096468.

Formula

a(n) = 2*A096033(n+2) (conjectured).
Showing 1-7 of 7 results.

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