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 1-6 of 6 results.

A020888 Ordered set of (a + b - c)/2 as (a,b,c) runs through all primitive Pythagorean triples with a

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 33, 33, 33, 33, 34, 34, 35, 35, 35, 35
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

n appears A068068(n) number of times. - Lekraj Beedassy, May 03 2006
Ordered inradii of primitive Pythagorean triangles. - Lekraj Beedassy, May 08 2006

Links

Crossrefs

For values ordered by hypotenuse, see A014498.

Formula

a(n) = A020887(n)/2.

Extensions

Offset corrected to 1 by Ray Chandler, Jan 23 2020

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

A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

Original entry on oeis.org

8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338
Offset: 1

Views

Author

Frank M Jackson, Oct 19 2016

Keywords

Comments

The mapping of the ordered pair (x,y) to an integer uses Cantor's pairing function to generate the integer as (x+y)(x+y+1)/2+y. Also for every ordered pair (x,y) such that 0 < x < y, gcd(x,y)=1 and x+y odd, there exists a primitive Pythagorean triple (PPT) (a, b, c) such that a = y^2-x^2, b = 2xy, c = x^2+y^2. Therefore each term in the sequence represents a unique PPT.
Numbers n for which 0 < A025581(n) < A002262(n) and A025581(n)+A002262(n) is odd, and gcd(A025581(n), A002262(n)) = 1. [The definition expressed with A-numbers.] - Antti Karttunen, Nov 02 2016
See also the triangle T(y, x) with the values for PPTs given in A278147. - Wolfdieter Lang, Nov 24 2016

Examples

			a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - _Antti Karttunen_, Nov 02 2016

Links

Crossrefs

Cf. A001477, A002262, A025581, A243808, A277632.
Cf. A020882 (is obtained when A048147(a(n)) is sorted into ascending order), A008846 (same with duplicates removed).
Cf. A020887, A020888, A120427, A024362, A024406, A046079, A046087, A070151, A156678, A156679, A156680, A156683, A156685, A222946, A278147.

Programs

  • Mathematica
    Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]

A087459 Values (X + Y - Z) sorted on Z, then on Y, where (X,Y,Z) is a primitive Pythagorean triple with X

Original entry on oeis.org

2, 4, 6, 6, 12, 10, 8, 20, 10, 24, 14, 30, 28, 12, 30, 40, 18, 42, 14, 36, 56, 22, 16, 42, 60, 70, 44, 18, 72, 48, 70, 26, 84, 66, 90, 20, 52, 80, 88, 30, 78, 22, 60, 90, 110, 112, 60, 126, 104, 24, 66, 132, 34, 126, 130, 144, 68, 26, 154, 120, 110, 140, 156, 38, 102, 28
Offset: 1

Views

Author

Lekraj Beedassy, Oct 23 2003

Keywords

Links

Crossrefs

For ordered values of (X + Y - Z) see A020887.
Cf. A046086, A046087, A020882, A014498.

Formula

a(n) = A046086(n) + A046087(n) - A020882(n) = 2*A014498(n).
a(n) = sqrt{2*A118961(n)*A118962(n)}. - Lekraj Beedassy, May 11 2006

Extensions

Corrected and extended by Ray Chandler, Oct 25 2003

A087484 Least hypotenuse of a primitive Pythagorean triangle with inradius n.

Original entry on oeis.org

5, 13, 17, 41, 37, 29, 65, 145, 101, 53, 145, 65, 197, 85, 73, 545, 325, 125, 401, 97, 109, 173, 577, 185, 677, 229, 785, 137, 901, 157, 1025, 2113, 205, 365, 169, 185, 1445, 445, 265, 233, 1765, 205, 1937, 241, 221, 629, 2305, 617, 2501, 733, 409, 305, 2917, 845
Offset: 1

Views

Author

Lekraj Beedassy, Oct 23 2003

Keywords

Comments

Least hypotenuse which is 2n short of the sum of the legs of a primitive Pythagorean triangle.

Links

Crossrefs

Cf. A020887, A099776 (largest hypotenuse).

Extensions

More terms from Ray Chandler, Oct 25 2003
Name simplified by Ray Chandler, Jan 26 2020
Deleted a link to a bad web site - N. J. A. Sloane, Jan 26 2020

A156686 The ordered set of a + b - c as (a,b,c) runs through all Pythagorean triples with a

Original entry on oeis.org

2, 4, 4, 6, 6, 6, 8, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 18, 20, 20, 20, 20, 20, 20, 22, 22, 22, 24, 24, 24, 24, 24, 24, 24, 24, 24, 26, 26, 26, 28, 28, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 30, 30, 30
Offset: 1

Views

Author

Ant King, Feb 18 2009

Keywords

Comments

Also called the excess of a Pythagorean triangle, and is equal to the diameter of its incircle. All members of this sequence are even, and the corresponding sequence for primitive triangles only is A020887.

Examples

			The smallest excess in any Pythagorean triangle is 2, which occurs in (3,4,5) because 3+4-5=2. Hence a(1)=2.

Links

Crossrefs

Cf. A020887.

Programs

  • Mathematica
    data1=Reduce[ a^2+b^2==c^2 && a+b-c==# && 0
Showing 1-6 of 6 results.

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

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