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A307894
Hypotenuses of primitive Pythagorean triangles with prime length, having the property that the sum and absolute difference of the shorter legs are both prime numbers.
0
13, 17, 37, 53, 73, 97, 109, 113, 137, 149, 193, 197, 233, 277, 317, 337, 401, 449, 457, 541, 613, 641, 653, 673, 709, 757, 809, 821, 877, 1009, 1061, 1093, 1117, 1129, 1201, 1289, 1297, 1381, 1481, 1549, 1733, 1873, 1877, 1913, 1933, 1997, 2017, 2053, 2153, 2213, 2221, 2377, 2417, 2437, 2557, 2797
OFFSET
1,1
COMMENTS
Replacing the shorter legs with the sum and absolute difference of the shorter legs may result in an integer-sided triangle, but this is not always the case. For example, {5,12,13}->{7,13,17} and {7,13,17} are the sides of a triangle. However, {60,91,109}->{31,109,151}, but {31,109,151} are not the sides of a triangle. If the replacement results in such a triangle, then the triangle is a scalene integer triangle (A070112) with sides of prime length, and a(n) is a term of A070081.
Sequence provides x-value of solutions to the equation 2*x^2 = y^2 + z^2, with x, y and z primes. - Lamine Ngom, Apr 30 2022
EXAMPLE
13 is a term because 12 + 5 = 17 and 12 - 5 = 7.
17 is a term because 15 + 8 = 23 and 15 - 8 = 7.
37 is a term because 35 + 12 = 47 and 35 - 12 = 23.
CROSSREFS
Subset of A008846.
Subset of A307880.
Sequence in context: A263725 A340053 A174056 * A322472 A166681 A109308
KEYWORD
nonn
AUTHOR
Torlach Rush, May 03 2019
STATUS
approved