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.

A118905 Sum of legs of Pythagorean triangles (without multiple entries).

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 102, 103, 105, 112, 113, 115, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 147, 151, 153, 154, 155, 158, 161, 164, 167, 168, 170, 175, 178, 182, 184, 186, 187, 188
Offset: 1

Views

Author

Giovanni Resta, May 05 2006

Keywords

Comments

The prime numbers in this sequence define A001132 (see comment in A001132). - Richard Choulet, Dec 16 2008
For the sum of legs of Pythagorean triangles with multiple entries see A198390. - Wolfdieter Lang, May 24 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
For the sum of legs of primitive Pythagorean triangles see A120681. - Wolfdieter Lang, Feb 17 2015
n is in the sequence iff A331671(n) > 0. - Ray Chandler, Feb 26 2020

Examples

			7 = 3 + 4 and 3^2 + 4^2 = 5^2.
a(14) = 49 = 7^2 from the primitive Pythagorean triangle (x,y,z) = (9,40,41), and from the non-primitive one 7*(3,4,5); a(42) = 119 = 7*17 from four Pythagorean triangles (39,80,89) and (99,20,181) (both primitive) and 7*(5,12,13), 17*(3,4,5). - _Wolfdieter Lang_, May 24 2013

Links

Crossrefs

Cf. A009096, A118903, A118904, A058529, A001132, A120681, A331671.

Programs

  • Magma
    [m:m in [2..200]|#[k:k in [1..m-1]|IsSquare(k^2+(m-k)^2)] ne 0]; // Marius A. Burtea, Jul 29 2019
  • PARI
    is(n)=my(t=n^2); forstep(i=2-n%2, n-2, 2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

Extensions

More terms from 147 on. - Richard Choulet, Nov 24 2009
Name specified. - Wolfdieter Lang, May 24 2013

A380436 Integers with at least 1 proper factorization for which the sum of the squares of the factors is a square.

Original entry on oeis.org

12, 16, 32, 36, 48, 60, 80, 81, 96, 108, 112, 120, 128, 140, 144, 168, 192, 220, 224, 240, 252, 256, 288, 300, 320, 336, 351, 360, 364, 396, 400, 420, 432, 448, 480, 486, 500, 512, 528, 540, 544, 560, 576, 588, 608, 612, 624, 625, 640, 644, 648, 660, 672, 704
Offset: 1

Views

Author

Charles L. Hohn, Jan 24 2025

Keywords

Examples

			a(1) = 12: {3, 4} (3 * 4 = 12 and 3^2 + 4^2 = 5^2).
a(2) = 16: {2, 2, 2, 2} (2 * 2 * 2 * 2 = 16 and 2^2 + 2^2 + 2^2 + 2^2 = 4^2).
a(3) = 32: {2, 4, 4} (2 * 4 * 4 = 32 and 2^2 + 4^2 + 4^2 = 6^2).

Links

Crossrefs

A118903 is a subsequence.

Programs

  • PARI
    a380436_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(issquare(sum(i=1, #f, f[i]^2)), 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dCharles L. Hohn, Mar 09 2025

A118904 Perimeters of rectangles with integer sides and diagonal.

Original entry on oeis.org

14, 28, 34, 42, 46, 56, 62, 68, 70, 82, 84, 92, 94, 98, 102, 112, 124, 126, 136, 138, 140, 142, 146, 154, 158, 164, 168, 170, 178, 182, 184, 186, 188, 194, 196, 204, 206, 210, 224, 226, 230, 238, 246, 248, 252, 254, 266, 272, 274, 276, 280, 282, 284, 292, 294
Offset: 1

Views

Author

Giovanni Resta, May 05 2006

Keywords

Examples

			14 = 2*(3+4) and 3^2+4^2=5^2.

Links

Crossrefs

Cf. A009096, A118902, A118903, A118905.

Formula

Twice A118905.

A181925 Area A of the triangles such that A, the sides, and at least one of the three bisectors are integers.

Original entry on oeis.org

12, 48, 60, 108, 120, 168, 192, 240, 300, 360, 420, 432, 480, 540, 588, 660, 672, 768, 960, 972, 1008, 1080, 1092, 1200, 1260, 1344, 1440, 1452, 1500, 1512, 1680, 1728, 1848, 1920, 1980, 2028, 2160, 2352, 2448, 2520, 2640, 2688, 2700, 2772, 2940, 3000
Offset: 1

Views

Author

Michel Lagneau, Apr 02 2012

Keywords

Comments

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.
The lengths of the bisectors are given by:
b1 = sqrt(bc*(b+c-a)(a+b+c))/(b+c)
b2 = sqrt(ac*(a+c-b)(a+b+c))/(a+c)
b3 = sqrt(ab*(a+b-c)(a+b+c))/(a+b)
Properties of this sequence: There exist three subsets of numbers included in a(n):
Case (i): A subset with a majority of isosceles triangles whose area equals the sum of the areas of two Pythagorean triangles with integer sides => the sequence A118903 is included in this sequence. This sort of triangles contains generally only one integer bisector, but more rarely three integer bisectors (see the examples).
Case (ii): Right triangles (a,b,c) where a^2 + b^2 = c^2.
Case (iii): A class of non-isosceles and non-right triangles (a, b, c) where one, two or three bisectors are integers.

Examples

			Case (i): 12 is in the sequence because the area of the isosceles triangle (5, 5, 6) equals 12 and one of the bisectors is an integer (4). But the isosceles triangle (546, 975, 975) whose area equals 255528 contains three integer bisectors: 936, 560, 560.
Case (ii): The right triangle (28, 96, 100) => A = 1344, and the integer median is m = 35.
Case (iii): The triangle (31091676, 46267375, 62553491) => A =  690494511777840, and the three bisectors are 51555075, 38342304 and 22314600.

References

  • Ralph H. Buchholz, On triangles with rational altitudes, angles bisectors or medians, PhD Thesis, University of Newcastle, Nov 1989.

Links

Crossrefs

Cf. A118903, A188158.

Programs

  • Maple
    with(numtheory):T:=array(1..1000):k:=0:nn:=300:for a from 1 to nn do: for b from a to nn do: for c from b to nn do:p:=(a+b+c)/2:s:=p*(p-a)*(p-b)*(p-c):aa:=b*c*(b+c-a)*(a+b+c): bb:=a*c*(a+c-b)*(a+b+c): cc:=a*b*(a+b-c)*(a+b+c):if s>0 and aa>0 and bb>0 and cc>0 then s1:=sqrt(s): aa1:=sqrt(aa)/(b+c): bb1:=sqrt(bb)/(a+c): cc1:=sqrt(cc)/(a+b):if s1=floor(s1) and (aa1=floor(aa1) or bb1=floor(bb1) or cc1=floor(cc1))  then k:=k+1:T[k]:=s1:else fi:fi:od:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):
  • Mathematica
    nn=300; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); aa=b*c*(b+c-a)*(a+b+c); bb=a*c*(a+c-b)*(a+b+c); cc=a*b*(a+b-c)*(a+b+c); If[0 < area2 && aa > 0 && bb > 0 && cc > 0 && IntegerQ[Sqrt[area2]] && (IntegerQ[Sqrt[aa]/(b+c)] || IntegerQ[Sqrt[bb]/(a+c)] || IntegerQ[Sqrt[cc]/(a+b)]), AppendTo[lst, Sqrt[area2]]]], {a,nn}, {b,a}, {c,b}]; Union[lst]
Showing 1-4 of 4 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.