A055193 -id:A055193 - cp's OEIS frontend

A094805 Duplicate of A055193.

6, 210, 840, 341880, 71831760, 64648584000, 2216650756320

Offset: 1

dead

A009127 Area of more than one Pythagorean triangle.

210, 840, 1320, 1890, 2730, 3360, 4914, 5250, 5280, 7560, 7980, 10290, 10920, 11880, 13440, 17010, 18480, 19656, 21000, 21120, 24570, 25410, 29400, 30240, 30600, 31920, 32130, 33000, 34650, 35490, 41160, 41580, 43680, 44226, 45144, 47250, 47520, 50490

Offset: 1

David W. Wilson

nonn

Duplicate numbers in A009111. The number 210 appears twice; 840 appears three times; 341880 appears four times. See A055193 for the sequence - T. D. Noe, Oct 14 2013

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

Cf. A009111.

A177063 Number of Pythagorean triangles with area n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

M. F. Hasler, Dec 09 2010

nonn

The first term > 1 is a(210) = 2, cf. A009127, A055193 and A024407. Up to there the sequence coincides with the characteristic function of A009112. The triangles are not necessarily primitive.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries related to Pythagorean triples.

Cf. A009111, A009127, A177021.

a(n)=my(N=1+#n=divisors(2*n));sum(i=1,N\2,issquare(n[i]^2+n[N-i]^2));

Secondary offset added by Antti Karttunen, Nov 24 2017

A093536 Least area common to n distinct primitive Pythagorean triangles.

Original entry on oeis.org

6, 210, 13123110

Offset: 1

Lekraj Beedassy, May 14 2004

a(4) >= 10^17. - Peter T. C. Radden, Dec 30 2012

a(4) >= 9.3*10^24, found by referencing A283274. - Frank A. Stevenson, Nov 26 2023

			a(2) = 210 is the smallest area common to 2 primitive Pythagorean triangles (viz. triples {20, 21, 29}, {12, 35, 37}); followed by 2730 (triples {60, 91, 109}, {28, 195, 197}); 7980 (triples {95, 168, 193}, {40, 399, 401}); 71610 (triples {341, 420, 541}, {132, 1085, 1093}); ...

Cf. A024407, A055193, A283274.

Corrected by Ray Chandler, Jul 02 2004

a(0) removed by Peter T. C. Radden, Dec 30 2012

A291591 Numbers k such that there exist exactly five distinct Pythagorean triangles, at least one of them primitive, with area k.

Original entry on oeis.org

71831760, 73513440, 1675212000, 6913932480, 4323749790360, 2678930100000, 175434192299520, 503151375767040

Offset: 1

Sture Sjöstedt, Aug 27 2017

I solve x^2 + 3*y^2 = (2*r)^2 over the positive integers. q, r, q-p and p are the y-coordinates in the first quadrant. Area = q*r*(q-p)*p. There are three Pythagorean triangles with this area. j, x, y with x > y and Area = j^2*x*y*(x-y)*(x+y) gives the area of an Pythagorean triangle.
Example: r = 169 in x^2 + 3*y^2 = (2*169)^2 gives q = 176, r = 169, q-p = 161 and p = 15;
k = q*r*(q-p)*p = 176*169*161*15 = 71831760.
j = 26, x = 23, y = 12 and j = 26, x = 28, y = 5 gives two Pythagorean triangles with k = 71831760;
k = 676*23*12*11*35 = 71831760 and k = 676*28*5*23*33 = 71831760.

			p^2 - p*q + q^2 = r^2;
p = 115, q = 448, q-p = 333, r = 403;
k = p*q*(q-p)*r = 115*448*333*403 = 6913932480.
x = 414, y = 104 and x = 558, y = 40 gives the same area.
k = x*y*(x-y)*(x+y) = 414*104*310*518 = 6913932480.
k = x*y*(x-y)*(x+y) = 558*40*518*598 = 6913932480.

Cf. A055193.

a(2), a(4)-a(7) from Giovanni Resta, Aug 28 2017

Missing term 73513440 inserted by Miguel-Ángel Pérez García-Ortega, Jul 19 2021

A055184 Number of new numbers in n-th segment of A055177; see example line of A055177.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 3, 3, 1, 2, 2, 2, 2, 3, 4, 2, 3, 4, 2, 4, 2, 5, 2, 1, 3, 6, 2, 4, 3, 4

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Also, the number of new numbers in n-th segment of A055193.

A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

Original entry on oeis.org

341880, 8168160, 14636160, 17957940, 52492440, 116396280, 1071572040, 1187525640, 1728483120, 5988702720, 6609482880, 22539095040, 29239970760, 136496680320, 258670630680, 398648544840, 494892478080, 592003418160, 1329673884000, 1343798407560, 2190884461920

Offset: 1

Sture Sjöstedt, Aug 23 2017

nonn

Numbers n such that there exist positive integers x, y with x > y and n = x*y*(x-y)*(x+y).

Many of them consist of a Pythagorean triangle plus a triple which is a solution to Carroll's problem: Find three Pythagorean triangles with the same area.

			p^2 - p*q + q^2 = r^2;
p = 208, q = 418, r = 362, q - p = 210;
n = p*r*q*(q-p) = 208*418*362*210 = 6609482880.
x = 640, y = 627 gives the same area:
n = x*y*(x-y)*(x+y) = 640*627*13*1267 = 6609482880.

Cf. A009127, A024407, A055193, A088513, A088977, A089025, A177021, A291591.

a(12)-a(21) from Giovanni Resta, Aug 28 2017

cp's OEIS Frontend

A094805 Duplicate of A055193.

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A009127 Area of more than one Pythagorean triangle.

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A177063 Number of Pythagorean triangles with area n.

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A093536 Least area common to n distinct primitive Pythagorean triangles.

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A291591 Numbers k such that there exist exactly five distinct Pythagorean triangles, at least one of them primitive, with area k.

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A055184 Number of new numbers in n-th segment of A055177; see example line of A055177.

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A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

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