A389589 - OEIS

A389589

Semiprimes that are (perimeter^2 - hypotenuse^2) of a Pythagorean triple.

119, 731, 13067, 13703, 31979, 44831, 45347, 76319, 90131, 137591, 137903, 164603, 207659, 362231, 416771, 503579, 579659, 872051, 891911, 961859, 1211831, 1418639, 1512887, 1823519, 1929107, 2336591, 2419031, 2599151, 2602211, 2760623, 2943251, 3124799, 3409919, 3868511, 3950831, 4231163

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers (x^2 + 2*x*y - y^2)*(3*x^2 + 2*x*y + y^2) where x and y are coprime with 0 < y < x and x^2 + 2*x*y - y^2 and 3*x^2 + 2*x*y + y^2 are both prime.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

a(3) = 13067 is a term because 13067 = (28+45) * (28+45+2*53) where (28, 45, 53) is a Pythagorean triple with 28+45 = 73 and 28+45+2*53 = 179 both prime.

MAPLE

N:= 10^8: # for terms <= N

S:= {}:

for x from 1 while 3*x^4 < N do

for y from 1 to x-1 do

if igcd(x, y) > 1 then next fi;

s:= (x^2 + 2*x*y - y^2)*(3*x^2 + 2*x*y + y^2);

p1:= x^2 + 2*x*y - y^2; if not isprime(p1) then next fi;

p2:= 3*x^2 + 2*x*y + y^2; if not isprime(p2) then next fi;

if p1*p2 <= N then S:= S union {p1*p2} fi

od od:

sort(convert(S, list));