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A177021 Numbers which are the area of exactly three Pythagorean triangles. 4
840, 3360, 7560, 10920, 13440, 21000, 30240, 31920, 41160, 43680, 53760, 68040, 84000, 98280, 101640, 120960, 127680, 141960, 164640, 166320, 174720, 189000, 215040, 242760, 272160, 273000, 286440, 287280, 303240, 336000, 370440, 393120, 406560, 444360 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The triangles need not be primitive. Number of terms less than 10^n: 0, 0, 1, 3, 14, 53, ....

13123110 is the smallest number which is the area of three primitive Pythagorean triangles, (1380,19019,19069)(3059,8580,9109) and (4485,5852,7373); this triple was found by Charles L. Shedd in 1945.

From Sture Sjöstedt, Dec 06 2016: (Start)

840 = 3*5*7*8; p=3, q=8, q-p=5, r=7 is a solution to p^2 - pq + q^2 = r^2. If r is a prime number in the sequence 7, 13, 19, ..., there are three Pythagorean triangles with the same area and at least one of them is primitive.

10920 = 7*8*13*15; p=7, q=15, q-p=8, r=13.

x^2 + 3*y^2 = 4*r^2 where r is a prime number in the sequence 7, 13, 19, ... gives lattice points that can be used to find solutions to p^2 - pq + q^2 = r^2. p, q, (q-p) and r are the y-coordinates in the first quadrant. (End)

REFERENCES

Morton Cohen, Charles Lutwidge Dodgson (Lewis Carroll), b. Jan. 27, 1832, d. Jan. 14, 1898, A Brief Biography, Vintage Books, ISBN 978-0-679-74562-4 (26 November 1996).

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Index entries related to Pythagorean Triples.

FORMULA

A177021 = { n | A177063(n)=3 }. - M. F. Hasler, Dec 09 2010

EXAMPLE

a(1) = 840 is the area of {15,112,113}, {24,70,74} & {40,42,58}.

a(2) = 3360 is the area of {30,224,226}, {48,140,148} & {80,84,116}.

a(3) = 7560 is the area of {45,336,339}, {72,210,222} & {120,126,174}.

MATHEMATICA

lst = {}; m = 2; While[ m < 10^3, n = 1; While[ n < m, If[ IntegerQ@ Sqrt[ m^2 + n^2], a = m*n/2; If[a < 10^6, AppendTo[ lst, a], n = m]]; n++ ]; m++ ]; Union@ Flatten@ Select[ Split@ Sort@ lst, Length@ # == 3 &]

CROSSREFS

Cf. A009112, A177063.

Sequence in context: A232099 A005952 A260473 * A276161 A135038 A215230

Adjacent sequences:  A177018 A177019 A177020 * A177022 A177023 A177024

KEYWORD

nonn

AUTHOR

Claudio Meller, on a suggestion by Antonio Roldán, Dec 08 2010

EXTENSIONS

Extended and edited by Robert G. Wilson v, Dec 08 2010

a(28)-a(34) from Giovanni Resta, Aug 16 2017

STATUS

approved

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Last modified November 19 07:19 EST 2018. Contains 317333 sequences. (Running on oeis4.)