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.

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A376430 Numbers that can appear as both short and long legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.

Original entry on oeis.org

12, 15, 35, 40, 45, 60, 72, 80, 105, 112, 132, 140, 165, 168, 180, 195, 209, 221, 231, 252, 255, 260, 275, 285, 299, 312, 325, 340, 380, 391, 399, 408, 420, 425, 440, 459, 465, 520, 532, 575, 595, 600, 609, 612, 651, 660, 700, 728, 741, 748, 759, 760, 779, 780, 800
Offset: 1

Views

Author

Hugo Pfoertner, Sep 23 2024

Keywords

Comments

Intersection of A002365 and A002366.

Examples

			a(1) = 12, because it is the long leg in the triangle (5, 12, 13) and the short leg in (12, 35, 37);
a(2) = 15: long leg in (8, 15, 17), short leg in (15, 112, 113).

Crossrefs

Cf. A002144, A002365, A002366, A376428, A376429.

A068386 One-thirtieth the area of the unique Pythagorean triangle whose hypotenuse is A002144(n), the n-th prime of the form 4k+1.

Original entry on oeis.org

1, 2, 7, 7, 6, 21, 11, 44, 52, 78, 33, 91, 28, 154, 119, 187, 143, 57, 266, 91, 221, 364, 418, 136, 299, 483, 616, 323, 130, 385, 840, 897, 1020, 1155, 1071, 1235, 266, 782, 203, 986, 1638, 1190, 1653, 1683, 2046, 2387, 1463, 2002, 460, 2852, 2204, 357
Offset: 2

Views

Author

Lekraj Beedassy, Mar 08 2002

Keywords

Comments

Every such prime p has a unique representation as p = r^2 + s^2 with 1 <= r < s. The corresponding right triangle has legs of lengths s^2 - r^2 and 2rs and area rs(s^2 - r^2). For p > 5, this is divisible by 30.
Calling A002330(n) and A002331(n) respectively u and v, we have a(n) = u*v*(u-v)*(u+v), for n > 1. - Lekraj Beedassy, Mar 12 2002
The corresponding Pythagorean triple (A, B, C) with A^2 = B^2 + C^2, (A > B > C) is given by {A002144(n), A002365(n), A002366(n)}, so that a(n) = B*C/(2*30) = A002365(n)*A002366(n)/60. - Lekraj Beedassy, Oct 27 2003

Examples

			The 7th prime of the form 4k+1 is 53 = 2^2 + 7^2. So the right triangle has sides 7^2 - 2^2 = 45, 2*2*7 = 28 and 53. Its area is 1/2 * 45 * 28 = 630, so a(7) = 630/30 = 21.

Crossrefs

Cf. A008846, A002144.

Programs

  • Mathematica
    a30[p_] := For[r=1, True, r++, If[IntegerQ[s=Sqrt[p-r^2]], Return[r s(s^2-r^2)/30]]]; a30/@Select[Prime/@Range[4, 150], Mod[ #, 4]==1&]
areat[p_]:=Module[{c=Flatten[PowersRepresentations[p,2,2]],a,b},a= First[c];b= Last[c];((b^2-a^2)(2a b))/2]; areat[#]/30&/@Select[Prime[ Range[4,200]],IntegerQ[(#-1)/4]&] (* Harvey P. Dale, Jun 21 2011 *)

Extensions

Edited by Dean Hickerson, Mar 14 2002

A139794 Interleaved reading of catheti and hypotenuses of Gaussian triangles with prime number hypotenuse.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 8, 15, 17, 20, 21, 29, 12, 35, 37, 9, 40, 41, 28, 45, 53, 11, 60, 61, 48, 55, 73, 39, 80, 89, 65, 72, 97, 20, 99, 101, 60, 91, 109, 15, 112, 113, 88, 105, 137, 51, 140, 149, 85, 132, 157, 52, 165, 173, 19, 180, 181, 95, 168, 193, 28, 195, 197, 60, 221
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 13 2008

Keywords

Formula

a(3n+1)= A002366(n). a(3n+2)=A002365(n). a(3n+3)=A002144(n).

Extensions

Edited by R. J. Mathar, Jun 15 2008

A211176 Numbers n which are the hypotenuse of a Pythagorean triple with n' as a leg, where n' is the arithmetic derivative of n.

Original entry on oeis.org

125, 625, 23125, 142805, 210125, 371293, 7983625, 9370805, 25757525, 50062025, 120670225, 489766225, 881052625, 1471596725, 2307267625, 2489771125, 3145529225, 3474871553, 6975757441, 7977558641
Offset: 1

Views

Author

Paolo P. Lava, Feb 01 2013

Keywords

Comments

This sequence is a subsequence of A008846. - Ray Chandler, Jan 27 2017

Examples

			n = 23125, n' = 19125 and sqrt(n^2-n'^2) = 13000.

Crossrefs

Cf. A002144, A002365, A002366, A003415, A008846, A009003, A009004, A210503.

Programs

  • Maple
    with(numtheory); ListA211176:= proc(q)local a,n,p;
for n from 2 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
if n<>a and type(sqrt(n^2-a^2),integer) then print(n); fi;
od; end: ListA211176(10^9);

Formula

A002144(n)^A002365(n) and A002144(n)^A002366(n) are terms of the sequence for all n. - Ray Chandler, Jan 27 2017

Extensions

Name and Maple program corrected by Paolo P. Lava, Sep 30 2013
a(12)-a(16) from Donovan Johnson, Sep 30 2013
a(17)-a(18) from Ray Chandler, Jan 25 2017
a(19)-a(20) from Ray Chandler, Jan 27 2017
Previous Showing 11-14 of 14 results.

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