A070079 -id:A070079 - cp's OEIS frontend

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.

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.

A281505 Numbers of the form y^2 - x^2 such that x^2 + y^2 is a prime and 0 < x < y.

Original entry on oeis.org

3, 5, 9, 11, 15, 19, 21, 25, 29, 35, 39, 45, 49, 51, 55, 59, 61, 65, 69, 71, 75, 79, 85, 91, 95, 99, 101, 105, 115, 121, 129, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 189, 195, 199, 201, 205, 209, 215, 219, 221

Offset: 1

Thomas Ordowski, Jan 23 2017

nonn

What is the natural density of this set of these numbers?
There are 204 terms up to 10^3, 1849 up to 10^4, 16881 up to 10^5, 160194 up to 10^6, 1531730 up to 10^7, and 14766494 up to 10^8. - Charles R Greathouse IV, Jan 23 2017
Numbers of the form s*t where 0 < s < t and (s^2 + t^2)/2 is prime. - Robert Israel, Jan 23 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Sam Chow and Carl Pomerance, Triangles with prime hypotenuse, arXiv:1703.10953 [math.NT], 2017.
Cihan Sabuncu, Right-angled triangles with almost prime hypotenuse, arXiv:2401.16334 [math.NT], 2024. Mentions this sequence.

Cf. A002144, A048161 is a subsequence, A070079 contains the same numbers.

filter:= proc(n)
  ormap(s -> isprime((s^2 + (n/s)^2)/2), select(s -> s^2Robert Israel, Jan 23 2017

filter[n_] := AnyTrue[Select[Divisors[n], #^2 < n & ], PrimeQ[(#^2 + (n/#)^2)/2] & ];
Select[Range[1, 1000, 2], filter] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

list(lim)=my(v=List()); for(a=1,sqrtint(lim\=1), for(x=1,(lim-a^2)\2\a, if(isprime((x+a)^2+x^2), listput(v,(x+a)^2-x^2)))); Set(v) \\ Charles R Greathouse IV, Jan 23 2017

a(n) = n(log n)^c /(log log n)^O(1), where c = 1 - (1 + log log 2)/log 2 = 0.086... Cf. A027424. - Conjectured by Carl Pomerance, Jan 25 2017

More terms from Altug Alkan, Jan 23 2017

a(17)-a(50) from Charles R Greathouse IV, Jan 23 2017

A140385 Consecutive area triples (y^2-x^2,4xy,p=x^2+y^2) associated with the Pythagorean primes A002144.

Original entry on oeis.org

3, 8, 5, 5, 24, 13, 15, 16, 17, 21, 40, 29, 35, 24, 37, 9, 80, 41, 45, 56, 53, 11, 120, 61, 55, 96, 73, 39, 160, 89, 65, 144, 97, 99, 40, 101, 91, 120, 109, 15, 224, 113, 105, 176, 137, 51, 280, 149, 85, 264, 157, 165, 104, 173, 19, 360, 181, 95, 336, 193, 195, 56, 197, 221

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

Triples A070079(i), 8*A070151(i), A002144(i), i=1,2,3... in compound order.

			(3,8,5) followed by (5,24,13) followed by (15,16,17) ...

Edited by R. J. Mathar, Jun 16 2008

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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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A281505 Numbers of the form y^2 - x^2 such that x^2 + y^2 is a prime and 0 < x < y.

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A140385 Consecutive area triples (y^2-x^2,4xy,p=x^2+y^2) associated with the Pythagorean primes A002144.

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