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 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, May 02 2001

Keywords

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

Examples

			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.

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

Crossrefs

Cf. A060829, A061408, A061409.
Even entries of A024355. Ordered union of A081925 and A081935.

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).

Extensions

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

A061408 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both positive squares; list all such pairs (x,y) ordered by values of y; sequence gives y values.

Original entry on oeis.org

4, 6, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 24, 26, 28, 28, 30, 30, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 42, 44, 44, 46, 48, 48, 48, 48, 50, 52, 52, 54, 54, 56, 56, 56, 58, 60, 60, 60, 60, 62, 64, 64, 64, 66, 66, 68, 68, 70, 70, 72
Offset: 0

Views

Author

Jason Earls, May 01 2001

Keywords

Comments

Each even integer y >= 4 occurs A056924(y/2) times. - Robert Israel, Dec 10 2017

Examples

			Pairs are [5, 4], [10, 6], [17, 8], [26, 10], [13, 12], [37, 12], [50, 14], ... For example, 5-4 = 1^2, 5+4 = 3^2.

References

  • Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

Links

Crossrefs

Cf. A056924, A061409, A060829, A060830.

Programs

  • Maple
    seq(y $ nops(select(t -> (t^2 < y/2), numtheory:-divisors(y/2))), y=2..100,2); # Robert Israel, Dec 10 2017
  • Mathematica
    Table[Table[y, {Select[Divisors[y/2], #^2 < y/2&] // Length}], {y, 2, 100, 2}] // Flatten (* Jean-François Alcover, Feb 28 2019, after Robert Israel *)

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. - N. J. A. Sloane, May 02 2001

Extensions

Definition clarified by Robert Israel, Dec 10 2017

A060829 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 x values.

Original entry on oeis.org

5, 17, 13, 37, 65, 29, 101, 25, 145, 53, 197, 257, 85, 325, 41, 401, 125, 485, 73, 577, 173, 677, 65, 785, 61, 109, 229, 901, 1025, 293, 1157, 97, 1297, 365, 1445, 89, 1601, 85, 205, 445, 1765, 137, 1937, 533, 2117, 265, 2305, 629, 2501, 185, 2705, 733, 2917
Offset: 0

Views

Author

N. J. A. Sloane, May 02 2001

Keywords

Examples

			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.
a(41) = 1765 because A120427(41) = 84 and we have gcd(1765,84)=1 and 1765-84 = 41^2 and 1765+84 = 43^2. - _Sean A. Irvine_, Jan 01 2023

References

  • Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

Crossrefs

Cf. A060830, A061408, A061409, A120427.

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.

Extensions

a(41) onward corrected by Sean A. Irvine, Jan 01 2023

A179484 Primes p such that p+-180 are squares.

Original entry on oeis.org

181, 349, 2029, 8101
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 16 2010

Keywords

Comments

Sequences of this type searching for p+d=s^2, p-d=t^2 (here: d=180) are finite because the difference 2d=(s+t)(s-t) has a finite set of solutions -- found by scanning the divisors of 2d (see A060829, A061409). [R. J. Mathar, Jul 20 2010]

Examples

			181-180=1^2;181+180=19^2;349-180=13^2,349+180=23^2;2029-180=43^2,2029+180=47^2;8101-180=89^2,8101+180=91^2;

Programs

  • Mathematica
    q=180;Select[Table[Prime[n],{n,3*9!}],IntegerQ[Sqrt[ #-q]]&&IntegerQ[Sqrt[ #+q]]&]

Extensions

keyword:fini,full inserted - Zak Seidov and R. J. Mathar, Jul 19 2010
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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