A376208 -id:A376208 - cp's OEIS frontend

A375750 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an odd short leg.

1, 3, 6, 10, 15, 16, 21, 22, 24, 28, 36, 37, 39, 42, 45, 46, 48, 51, 55, 58, 66, 67, 69, 72, 76, 78, 79, 84, 88, 91, 94, 97, 105, 106, 108, 111, 115, 120, 121, 123, 126, 130, 135, 136, 139, 142, 153, 154, 157, 163, 168, 171, 172, 174, 177, 181, 186, 190, 192, 193

Offset: 1

Hugo Pfoertner, Sep 13 2024

nonn

Sorted distinct values of ({A081961} - 1)/4.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A008846, A081961, A376208.

\\ Uses function is_a376208 from A376208
is_a376208(n,1)

from math import gcd, isqrt
test_all_k_upto   = 193
A375750, x, limit = set(), 2, test_all_k_upto * 4 + 1
while x**2 + (lowY := isqrt(2*x**2)-x)**2 < limit:
   for y in range(min(x-1,(yy:=isqrt(limit - x**2))-(yy%2 == x%2)), lowY,-2):
       if gcd(x,y) == 1: A375750.add(((x**2 + y**2) - 1) // 4)
   x += 1
print(A375750:=sorted(A375750)) # Karl-Heinz Hofmann, Sep 17 2024

A376210 Numbers k for which among all possible Pythagorean triangles with the hypotenuse 4*k+1, the minimum of the lengths of the shorter legs is even.

Original entry on oeis.org

4, 7, 9, 13, 16, 18, 25, 27, 34, 43, 49, 57, 60, 64, 70, 73, 81, 87, 93, 99, 100, 102, 111, 112, 114, 121, 123, 127, 133, 144, 148, 150, 157, 160, 165, 169, 175, 183, 186, 189, 196, 202, 207, 211, 214, 219, 225, 235, 241, 244, 249, 255, 256, 258, 262, 265, 273

Offset: 1

Hugo Pfoertner, Sep 21 2024

nonn

			       Hypotenuses                A376210
       4k+1                       |  A376211
   k   A008846                    |  |  A376208
   |   |  Sorted legs [x,y] of    |  |  |  A375750
   |   |  Pythagorean triangles   |  |  |  |  A376209
   1   5  [3,4]                   .  X  .  X  .
   3  13  [5,12]                  .  X  .  X  .
   4  17  [8,15]                  X  .  X  .  .
   6  25  [7,24]                  .  X  .  X  .
   7  29  [20,21]                 X  .  X  .  .
   9  37  [12,35]                 X  .  X  .  .
  10  41  [9,40]                  .  X  .  X  .
  13  53  [28,45]                 X  .  X  .  .
  15  61  [11,60]                 .  X  .  X  .
  16  65  [16,63],[33,56],[39,52] X  .  X  X  X
  18  73  [48,55]                 X  .  X  .  .
  21  85  [13,84],[36,77],[51,68] .  X  X  X  X

({A087937}-1)/4 is a subsequence.
A094178 \ A376211.
Cf. A375750, A376208.

is_a376210_1(n,r=0) = my(c=4*n+1, q=qfbsolve(Qfb(1,0,1), c^2, 3), qd=#q); if(qd<2, 0, my(a=vecmin(abs(concat(q))[1..2*(qd-1)]), b=sqrtint(c^2-a^2)); a%2==r && gcd([a,b,c])==1)

A376209 Numbers k for which the hypotenuse z=4*k+1 occurs in more than one primitive Pythagorean triangle, such that 2 distinct triangles (x1,y1,z), (x2,y2,z) with opposite parity of their short legs x1 and x2 exist (xi < yi < z).

Original entry on oeis.org

16, 21, 36, 46, 51, 55, 66, 76, 91, 94, 111, 121, 123, 126, 136, 157, 171, 172, 181, 186, 196, 211, 216, 225, 237, 241, 246, 268, 276, 286, 289, 291, 297, 301, 310, 315, 331, 336, 346, 351, 354, 361, 366, 378, 384, 396, 412, 421, 436, 441, 442, 445, 456, 463, 466

Offset: 1

Hugo Pfoertner, Sep 21 2024

nonn

			See A376210.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Intersection of A375750 and A376208.

Cf. A376210, A376211.

\\ Uses function is_a376208 from A376208
is(n) = is_a376208(n,0) && is_a376208(n,1)

A376211 Numbers k for which among all possible Pythagorean triangles with the hypotenuse 4*k+1, the minimum of the lengths of the shorter legs is odd.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 22, 24, 28, 36, 37, 39, 45, 46, 48, 55, 58, 66, 67, 69, 78, 79, 84, 88, 91, 94, 97, 105, 108, 115, 120, 130, 135, 136, 139, 142, 153, 154, 163, 168, 171, 172, 177, 190, 192, 193, 199, 205, 210, 213, 220, 231, 232, 234, 237, 238, 252, 253, 267

Offset: 1

Hugo Pfoertner, Sep 21 2024

nonn

			See A376210.

({A087938}-1)/4 is a subsequence.
A094178 \ A376210.
Cf. A375750, A376208.

\\ uses function is_a376210_1 from A376210
is_a376210_1(n,1)

cp's OEIS Frontend

A375750 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an odd short leg.

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A376210 Numbers k for which among all possible Pythagorean triangles with the hypotenuse 4*k+1, the minimum of the lengths of the shorter legs is even.

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A376209 Numbers k for which the hypotenuse z=4*k+1 occurs in more than one primitive Pythagorean triangle, such that 2 distinct triangles (x1,y1,z), (x2,y2,z) with opposite parity of their short legs x1 and x2 exist (xi < yi < z).

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Programs

PARI

A376211 Numbers k for which among all possible Pythagorean triangles with the hypotenuse 4*k+1, the minimum of the lengths of the shorter legs is odd.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI