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A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime. 8
4, 12, 60, 180, 420, 1740, 1860, 2520, 3120, 5100, 8580, 9660, 16380, 19800, 36720, 60900, 71820, 83640, 100800, 106260, 135720, 161880, 163020, 199080, 205440, 218460, 273060, 282000, 337020, 388080, 431520, 491040, 531480, 539760, 552300 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Apart from the first two terms, every term is divisible by 60 and is of the form 450*k^2 +/- 30*k or 450*k^2 +/- 330*k + 60 for some k.
In such a triangle, this even leg is always the longer leg, and the hypotenuse = a(n) + 1. The Pythagorean triples are (A048161(n), a(n), A067756(n)), so, for a(2) = 12, the corresponding Pythagorean triple is (5, 12, 13). - Bernard Schott, Apr 12 2023
LINKS
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.
FORMULA
a(n) = (A048161(n)^2 - 1)/2 = A067756(n) - 1.
EXAMPLE
4 is a term: in the right triangle (3, 4, 5), 3 and 5 are prime.
5100 is a term: in the right triangle (101, 5100, 5101), 101 and 5101 are prime.
MATHEMATICA
lst={}; Do[q=(Prime[n]^2+1)/2; If[PrimeQ[q], AppendTo[lst, (Prime[n]^2-1)/2]], {n, 200}]; lst (* Frank M Jackson, Nov 02 2013 *)
CROSSREFS
Cf. A048161, A067756. Contains every value of A051858.
Sequence in context: A243923 A192331 A068525 * A051858 A084709 A057394
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jan 31 2002
STATUS
approved

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Last modified May 26 03:59 EDT 2024. Contains 372807 sequences. (Running on oeis4.)