A253802 -id:A253802 - cp's OEIS frontend

A253803 a(n) gives one fourth of the even leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253802(n).

6, 39, 60, 210, 210, 410, 630, 915, 1320, 1780, 2340, 990, 2730, 3164, 4620, 5215, 5610, 4290, 8145, 8106, 2730, 6630, 12116, 12540, 4080, 17485, 17451, 18480, 9690, 24414

Offset: 1

Wolfdieter Lang, Jan 14 2015

See A253802 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253802(7)^2 + (4*a(7))^2 = 1241^2 + (4*630)^2.
The other Pythagorean triangle with hypotenuse
53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4*A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A080109, A253802, A253804, A253805.

a(n) = sqrt(A080109(n)^2 - A253802(n)^2)/4, n >= 1.

A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

Original entry on oeis.org

15, 119, 255, 609, 1295, 1519, 2385, 3479, 4015, 4879, 6305, 9999, 9919, 12319, 14385, 16999, 13345, 28545, 32039, 19199, 38415, 50609, 32239, 50369, 65535, 62839, 50279, 64911, 83505, 96719

Offset: 1

Wolfdieter Lang, Jan 16 2015

The corresponding even legs are given in 4*A253805.
The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253802(n) (odd) and A253803(n) (even).
Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^= A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
Concerning the primitivity question of these triangles see a comment on A253802.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = a(7)^2 + (4*A253805(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253303, A253802, A253805.

A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253805(n))^2,
n >= 1, that is,
a(n) = sqrt(A080175(n) - (4*A253805(n))^2), n >= 1.

A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

Original entry on oeis.org

5, 30, 34, 145, 111, 180, 371, 330, 876, 1560, 1746, 505, 1635, 840, 3014, 3570, 5181, 2249, 1710, 7980, 1379, 3435, 10920, 7230, 2056, 8970, 14490, 11240, 4981, 3900

Offset: 1

Wolfdieter Lang, Jan 16 2015

See A253804 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253804(7)^2 + (4*a(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A080109, A253804, A253802, A253803.

a(n) = sqrt(A080109(n)^2 - A253804(n)^2)/4, n >= 1.

cp's OEIS Frontend

A253803 a(n) gives one fourth of the even leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253802(n).

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A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

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A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

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Author

Keywords

Comments

Examples

References

Crossrefs

Formula