A389663 Semiprimes that are (perimeter^3 - hypotenuse^3) of a Pythagorean triple.

1603, 24803, 3377639, 5414443, 48942259, 80019263, 99235051, 139774907, 150121999, 245836051, 1003279951, 1211176843, 5594769907, 5781637063, 8959411267, 9285738167, 15951900491, 16731856343, 18435176479, 21043491167, 23347922303, 29444639443, 31390478279, 72633432647, 91786264751, 91841783039

Offset

1,1

Comments

Numbers (x^2 + 2*x*y - y^2) * (7*x^4 + 10*x^3*y + 8*x^2*y^2 + 2*x*y^3 + y^4) where x and y are coprime with 0 < y < x and both x^2 + 2*x*y - y^2 and 7*x^4 + 10*x^3*y + 8*x^2*y^2 + 2*x*y^3 + y^4 are prime.

Links

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

Example

a(3) = 3377639 is a term because 3377639 = 89 * 37951 where 89 and 37951 are primes and 3377639 = 154^3 - 65^3 where (33, 56, 65) is a Pythagorean triple with perimeter 154 and hypotenuse 65.

Maple

N:= 10^11: # for terms <= N
S:= {}:
for x from 1 while 7*x^6 + 24*x^5 + 21*x^4 + 8*x^3 - 3*x^2 - 1 <= N do
  for y from 1 to x-1 do
     if igcd(x, y) > 1 then next fi;
     if (2*x^2 + 2*x*y)^3 - (x^2 + y^2)^3 > N then break fi;
     p1:= x^2 + 2*x*y - y^2;
     if not isprime(p1) then next fi;
     p2:= 7*x^4 + 10*x^3*y + 8*x^2*y^2 + 2*x*y^3 + y^4;
     if isprime(p2) then S:= S union {p1*p2} fi;
od od:
sort(convert(S, list));

Crossrefs

Cf. A103605, A389589.

Sequence in context: A293370 A352092 A252439 * A224949 A171466 A205267

Adjacent sequences: A389660 A389661 A389662 * A389664 A389665 A389666

Keyword

nonn

Author

Will Gosnell and Robert Israel, Jan 11 2026

Status

approved