A081934 -id:A081934 - cp's OEIS frontend

A088557 Least even leg of primitive Pythagorean triangles with odd leg 2n+1.

4, 12, 24, 40, 60, 84, 8, 144, 180, 20, 264, 312, 364, 420, 480, 56, 12, 684, 80, 840, 924, 28, 1104, 1200, 140, 1404, 48, 176, 1740, 1860, 16, 72, 2244, 260, 2520, 2664, 308, 36, 3120, 3280, 3444, 132, 416, 3960, 60, 476, 168, 4704, 20, 5100, 5304, 88, 5724, 5940

Offset: 1

Lekraj Beedassy, Nov 17 2003

nonn

a(n) shares identical entries with A046092(n) except for n=(A081934 - 1)/2.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A088558, A096891-A096900.

a[n_] := 2k /. Solve[k > 0 && 1 < h < 2k + 2n + 1 && (2k)^2 + (2n + 1)^2 == h^2 && GCD[2k, 2n + 1, h] == 1, {k, h}, Integers][[1]];
a /@ Range[54] (* Jean-François Alcover, Mar 05 2020 *)

Corrected and extended by Ray Chandler, Jul 14 2004

A120890 Ordered odd leg of primitive Pythagorean triangles.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 15, 17, 19, 21, 21, 23, 25, 27, 29, 31, 33, 33, 35, 35, 37, 39, 39, 41, 43, 45, 45, 47, 49, 51, 51, 53, 55, 55, 57, 57, 59, 61, 63, 63, 65, 65, 67, 69, 69, 71, 73, 75, 75, 77, 77, 79, 81, 83, 85, 85, 87, 87, 89, 91, 91, 93, 93, 95, 95, 97, 99, 99, 101, 103

Offset: 1

Lekraj Beedassy, Jul 12 2006

nonn

Ordered union of A081874 and A081934.

Conjecture: lim_{n->oo} a(n)/n = 1/Pi. Limit is also conjectured to be equal to lim_{n->oo} A120427(n)/n, see Selle reference, chapter 2.3.10. - Lothar Selle, Jun 21 2022

Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 4th impression 2022, chapter 2.2.1., see chapter 2.3.10 for identity of lim_{n->oo} A120427(n)/n.

Ray Chandler, Table of n, a(n) for n = 1..10000
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.
F. Richman, Pythagorean Triangles

Cf. A061346, A081874, A081934, A120891.

Cf. A120427.

Corrected by T. D. Noe, Oct 25 2006

A112398 Odd numbers which are exclusively the shortest side of primitive Pythagorean triangles.

Original entry on oeis.org

3, 7, 9, 11, 19, 23, 27, 31, 33, 39, 43, 47, 49, 51, 57, 59, 67, 69, 71, 75, 79, 81, 83, 87, 93, 95, 103, 107, 111, 115, 119, 121, 123, 127, 129, 131, 133, 135, 139, 141, 147, 151, 155, 159, 161, 163, 167, 175, 177, 179, 183, 189, 191, 199, 201, 203, 207, 211, 213

Offset: 1

Lekraj Beedassy, Dec 06 2005

nonn

Numbers that are in A081874 (odd short legs) but in neither A081934 (odd long legs) nor A020882 (hypotenuse).

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

Entries of A112372 excluding entries in A020882.

Extended by Ray Chandler, Dec 08 2005

A112372 Odd legs of primitive Pythagorean triangles which are exclusively the short one.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 145, 147, 149, 151

Offset: 1

Lekraj Beedassy, Dec 02 2005

nonn

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

Entries of A081874 (removing duplicates) excluding its subsequence A081934.

Cf. A112398.

Extended by Ray Chandler, Dec 08 2005

cp's OEIS Frontend

A088557 Least even leg of primitive Pythagorean triangles with odd leg 2n+1.

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A120890 Ordered odd leg of primitive Pythagorean triangles.

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A112398 Odd numbers which are exclusively the shortest side of primitive Pythagorean triangles.

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A112372 Odd legs of primitive Pythagorean triangles which are exclusively the short one.

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