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 1-5 of 5 results.

A060830 Erroneous version of A120427.

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, 88, 92, 96, 100, 104, 108, 112, 116, 120, 120, 120, 124, 132, 132, 132, 136, 140, 140, 140
Offset: 1

Views

Author

Keywords

A225952 Triangle read by rows, giving the even legs of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

Original entry on oeis.org

4, 0, 12, 8, 0, 24, 0, 20, 0, 40, 12, 0, 0, 0, 60, 0, 28, 0, 56, 0, 84, 16, 0, 48, 0, 80, 0, 112, 0, 36, 0, 72, 0, 0, 0, 144, 20, 0, 60, 0, 0, 0, 140, 0, 180, 0, 44, 0, 88, 0, 132, 0, 176, 0, 220, 24, 0, 0, 0, 120, 0, 168, 0, 0, 0, 264, 0, 52, 0, 104, 0, 156, 0, 208, 0, 260, 0, 312, 28, 0, 84, 0, 140, 0, 0, 0, 252, 0, 308, 0, 364
Offset: 2

Views

Author

Wolfdieter Lang, May 23 2013

Keywords

Comments

For primitive Pythagorean triples (x,y,z) see the Niven et al. reference, Theorem 5.5, p. 232, and the Hardy-Wright reference, Theorem 225, p. 190.
There is a one-to-one correspondence between the values n and m of this number triangle for which a(n,m) does not vanish and primitive solutions of x^2 + y^2 = z^2 with y even, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2. The mirror triangles with x even are not considered here. Therefore a(n,m) = 2*n*m (for these solutions). The number of non-vanishing entries in row n is A055034(n).
The sequence of the main diagonal is 2*n*(n-1) = 4*A000217 (n-1), n >= 2.
If the 0 entries are eliminated and the numbers are ordered nondecreasingly (multiple entries appear) the sequence becomes A120427. All its entries are positive integer multiples of 4, shown in A008586(n), n >= 1. Note that all even legs <= N are certainly reached if one considers in the triangle rows n = 2, ..., floor(N/2).

Examples

			The triangle a(n,m) begins:
n\m   1   2   3   4    5    6    7    8    9    10   11 ...
2:    4
3:    0  12
4:    8   0  24
5:    0  20   0  40
6:   12   0   0   0   60
7:    0  28   0  56    0   84
8:   16   0  48   0   80    0  112
9:    0  36   0  72    0    0    0  144
10:  20   0  60   0    0    0  140    0  180
11:   0  44   0  88    0  132    0  176    0   220
12:  24   0   0   0  120    0  168    0    0     0  264
...

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
  • Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.

Links

Crossrefs

Cf. A222946 (hypotenuses), A225950 (odd legs), A225949 (leg sums), A225951 (perimeters), A120427 (even legs ordered), A008586 (multiples of 4).

Formula

a(n,m) = 2*n*m if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^{n+m} = -1), otherwise a(n,m) = 0.

Extensions

Edited. Refs. added. - Wolfdieter Lang, Jul 26 2014

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

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

Views

Author

Lekraj Beedassy, Jul 12 2006

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

Cf. A061346, A081874, A081934, A120891.
Cf. A120427.

Extensions

Corrected by T. D. Noe, Oct 25 2006

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]}]
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.