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A165238 Hypotenuses c of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes. 2
29, 65, 293, 485, 785, 1049, 1469, 1961, 2105, 3005, 3725, 3821, 4145, 4181, 4685, 4745, 5105, 5501, 6053, 6929, 6953, 7121, 7361, 7841, 8693, 9029, 9125, 10025, 12041, 12833, 12965, 13649, 14285, 14909, 15173, 15689, 15773, 15821, 16493 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Only one copy of c enters the sequence if multiple solutions exist, like with (a,b,c) = (3164,12573,12965) and (a,b,c) = (483,12956,12965).
Subsequence of A020882. [R. J. Mathar, Mar 25 2010]
LINKS
Eric Weisstein, Pythagorean Triple, MathWorld
EXAMPLE
(a,b,c) = (20,21,29), (33,56,65), (44,483,485), (56,783,785), (68,285,293), (273,4136,4145).
In the first case, for example, 2*20+1=41, 2*21+1 and 2*29+1 are all prime, which adds the half-hypotenuse 29 to the sequence.
MATHEMATICA
amax=6*10^4; lst={}; k=0; q=12!; Do[If[(e=((n+1)^2-n^2))>amax, Break[]];
Do[If[GCD[m, n]==1, a=m^2-n^2; If[PrimeQ[2*a+1], b=2*m*n; If[PrimeQ[2*b+1],
If[GCD[a, b]==1, If[a>b, {a, b}={b, a}]; If[a>amax, Break[]]; c=m^2+n^2;
If[PrimeQ[2*c+1], k++; AppendTo[lst, c]]]]]]; If[a>amax, Break[]], {m, n+1, 12!, 2}], {n, 1, q, 1}]; Union@lst
CROSSREFS
Sequence in context: A044131 A044512 A211492 * A201022 A367151 A118481
KEYWORD
nonn
AUTHOR
EXTENSIONS
Comments moved to examples and definition clarified by R. J. Mathar, Mar 25 2010
STATUS
approved

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Last modified May 30 04:46 EDT 2024. Contains 372958 sequences. (Running on oeis4.)