A024355 -id:A024355 - cp's OEIS frontend

A120427 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives y values.

4, 8, 12, 12, 16, 20, 20, 24, 24, 28, 28, 32, 36, 36, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 60, 60, 60, 60, 64, 68, 68, 72, 72, 76, 76, 80, 80, 84, 84, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 104, 104, 108, 108, 112, 112, 116, 116, 120, 120, 120, 120, 124, 124, 128

Offset: 1

N. J. A. Sloane, May 02 2001

nonn

Ordered even legs of primitive Pythagorean triangles.
I wrote an arithmetic program once to find out if and when y 'catches up to' n in A120427 (ordered even legs of primitive Pythagorean triples). It's around 16700. As enumerated by the even - or odd - legs, (not sure about the hypotenuses), the triples are 'denser' than the integers. - Stephen Waldman, Jun 12 2007
Conjecture: lim_{n->oo} a(n)/n = 1/Pi. - Lothar Selle, Jun 19 2022

			Pairs are [5, 4], [17, 8], [13, 12], [37, 12], [65, 16], [29, 20], [101, 20], ... E.g., 5-4 = 1^2, 5+4 = 3^2.

Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 3rd impression 2022, chapter 2.3.1.
Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Cf. A060829, A061408, A061409.

Even entries of A024355. Ordered union of A081925 and A081935.

The solutions are given by x = r^2 + 2*r*k + 2*k^2, y = 2*k*(k+r) with r >= 1, k >= 1, r odd, gcd(r, k) = 1.

a(n) = 2*A020887(n) = 4*A020888(n).

Corrected by Lekraj Beedassy, Jul 12 2007 and by Stephen Waldman (brogine(AT)gmail.com), Jun 09 2007

A156683 Integers that can occur as either leg in more than one primitive Pythagorean triple.

Original entry on oeis.org

12, 15, 20, 21, 24, 28, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 65, 68, 69, 72, 75, 76, 77, 80, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 111, 112, 115, 116, 117, 119, 120, 123, 124, 129, 132, 133, 135, 136, 140, 141, 143, 144

Offset: 1

Ant King, Feb 17 2009

This is also the sequence of non-singly-even numbers that contain more than one distinct prime factor.

Integers n such that A024361(n)>1; subsequence of both A024355 and A042965. - Ray Chandler, Feb 03 2020

			As 15 is the second integer that can occur as either leg in more than one primitive Pythagorean triangle - (8,15,17) and (15,112,113) - then a(2)=15.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A024355, A024361, A042965.

PrimitiveRightTriangleLegs[1]:=0;PrimitiveRightTriangleLegs[n_Integer?Positive]:=Module[{f=Transpose[FactorInteger[n]][[1]]},If[Mod[n,4]==2,0,2^(Length[f]-1)]];Select[Range[150],PrimitiveRightTriangleLegs[ # ]>1 &]

is(n)=n%4!=2 && !isprimepower(n) && n>1 \\ Charles R Greathouse IV, Jun 17 2013

cp's OEIS Frontend

A120427 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives y values.

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