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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
REFERENCES
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.
LINKS
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.
FORMULA
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).
EXAMPLE
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.
CROSSREFS
Even entries of A024355. Ordered union of A081925 and A081935.
Sequence in context: A334718 A334755 A333597 * A060830 A080458 A147646
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 02 2001
EXTENSIONS
Corrected by Lekraj Beedassy, Jul 12 2007 and by Stephen Waldman (brogine(AT)gmail.com), Jun 09 2007
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)