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Revision History for A343769 (Underlined text is an addition; strikethrough text is a deletion.)

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A343769 Sorted areas of primitive Heronian triangles for which a rectangle exists with integer dimensions and with perimeter and area equal respectively to the perimeter and area of the triangle.
(history; published version)
#8 by N. J. A. Sloane at Sat Oct 09 07:41:54 EDT 2021
STATUS

proposed

approved

#7 by Michel Marcus at Thu Sep 09 13:47:14 EDT 2021
STATUS

editing

proposed

#6 by Michel Marcus at Thu Sep 09 13:47:08 EDT 2021
EXAMPLE

a(1) = 12 because 12 is the area of the 5-5-6 triangle, which is the least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (2-by-6) and with perimeter (16) and area (12) equal respectively to the perimeter and area of the triangle. a(2) = 126 because 126 is the area of the 13-20-21 triangle, which is the second-least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (6-by-21) and with perimeter (54) and area (126) equal respectively to the perimeter and area of the triangle.

a(2) = 126 because 126 is the area of the 13-20-21 triangle, which is the second-least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (6-by-21) and with perimeter (54) and area (126) equal respectively to the perimeter and area of the triangle.

KEYWORD

nonn,more,changed

STATUS

proposed

editing

#5 by Jason Zimba at Thu Sep 09 12:22:58 EDT 2021
STATUS

editing

proposed

Discussion
Thu Sep 09 13:45 
Michel Marcus: Well I was just wondering
#4 by Jason Zimba at Wed Sep 08 08:50:32 EDT 2021
EXAMPLE

a(1) = 12 because 12 is the area of the 5-5-6 triangle, which is the least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (2-by-6) and with perimeter (16) and area (12) equal respectively to the perimeter and area of the triangle. a(2) = 126 because 126 is the area of the 13-20-21 triangle, which is the second-least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (6-by-21) and with perimeter (54) and area (126) equal respectively to the perimeter and area of the triangle.

STATUS

proposed

editing

#3 by Jason Zimba at Wed Sep 08 08:48:52 EDT 2021
STATUS

editing

proposed

#2 by Jason Zimba at Wed Apr 28 17:56:39 EDT 2021
NAME

allocatedSorted areas of primitive Heronian triangles for which a rectangle exists with integer dimensions and with perimeter and area equal respectively to the perimeter and area of Jasonthe Zimbatriangle.

DATA

12, 126, 624, 1260, 1800, 2100, 2850, 4536, 5292, 5580, 8820, 9900, 12600, 12642, 14850, 15600, 17640, 19110, 21756, 23400, 24948, 25200, 25536, 28350, 47040, 47304

OFFSET

1,1

LINKS

Jason Zimba, <a href="https://jzimba.blogspot.com/2021/04/there-are-infinitely-many-rectangular.html">There are infinitely many rectangular Heronian triangles</a>.

EXAMPLE

a(1) = 12 because 12 is the area of the 5-5-6 triangle, which is the least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (2-by-6) and with perimeter (16) and area (12) equal respectively to the perimeter and area of the triangle. a(2) = 126 because 126 is the area of the 13-20-21 triangle, which is the least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (6-by-21) and with perimeter (54) and area (126) equal respectively to the perimeter and area of the triangle.

MATHEMATICA

(* Adapted from Albert Lau's program for A224301 *)

AMax = 10000;

Do[c = p/b;

a1 = Sqrt[b^2 + c^2 + 2 Sqrt[b^2 c^2 - 4 A^2]];

a2 = Sqrt[b^2 + c^2 - 2 Sqrt[b^2 c^2 - 4 A^2]];

If[IntegerQ[a2] && GCD[a2, b, c] == 1 &&

a1 > a2 >= b && (per = a2 + b + c;

IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A, a2, b, c}//

Sow*)];

If[IntegerQ[a1] &&

GCD[a1, b, c] == 1 && (per = a1 + b + c;

IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A, a1, b, c}//

Sow*)]; , {A, 6, AMax, 6}, {p,

4 A^2 // Divisors //

Select[#, EvenQ[#] && # >= 2 A &] & // #/2 + 2 A^2/# & //

Select[#, IntegerQ] &}, {b,

p // Divisors // Select[#, #^2 >= p &] &}] // Reap // Last // Last

{a1, a2, c} =.;

CROSSREFS

Subsequence of A224301.

KEYWORD

allocated

nonn

AUTHOR

Jason Zimba, Apr 28 2021

STATUS

approved

editing

Discussion
Wed May 05 18:38 
OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
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Thanks.
  - The OEIS Server
Wed May 12 19:56 
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  - The OEIS Server
Wed May 19 22:47 
OEIS Server: This sequence has not been edited or commented on for a week
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Wed May 26 23:57 
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Thu Jun 03 01:06 
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Thu Aug 19 15:22 
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Thu Aug 26 16:46 
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Thu Sep 02 17:42 
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Wed Sep 08 08:33 
Michel Marcus: why don't you submit your sequence ???
08:48 
Jason Zimba: Well I drafted the sequence and was going to submit it for review, but then I thought I should wait until I had obtained some form of peer review first. I have a problem upcoming in the College Math Journal that I thought would be a good citation. In the meantime, I get a weekly reminder from the OEIS server, which I simply ignore. But I apologize. I had assumed that this process was confined only to me plus 1 robot...did not realize that editors would be having to deal with it. I'll go ahead and submit it. Best, Jason
#1 by Jason Zimba at Wed Apr 28 17:56:39 EDT 2021
NAME

allocated for Jason Zimba

KEYWORD

allocated

STATUS

approved

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Last modified August 31 00:13 EDT 2024. Contains 375550 sequences. (Running on oeis4.)