A390249 Numbers k that are the hypotenuse of at least one primitive Pythagorean triangle whose perimeter + inradius and perimeter - inradius are both prime.

5, 17, 25, 37, 61, 65, 89, 101, 109, 169, 193, 205, 221, 265, 281, 365, 373, 377, 401, 481, 485, 493, 533, 601, 653, 689, 709, 757, 797, 845, 865, 901, 905, 925, 965, 1025, 1037, 1073, 1117, 1165, 1313, 1417, 1445, 1453, 1493, 1517, 1553, 1597, 1657, 1685, 1769, 1885, 1937, 1945, 2041, 2069, 2117

Offset

1,1

Comments

Numbers of the form m^2 + n^2 where 1 <= n < m, m and n are coprime with one odd and one even, and 2*m^2 + m*n + n^2 and 2*m^2 + 3*m*n - n^2 are prime.

Links

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

Example

a(3) = 25 is a term because 25 is the hypotenuse of the primitive Pythagorean triangle with sides 24, 7, 25 having perimeter 24 + 7 + 25 = 56 and inradius (24 + 7 - 25)/2 = 3 and both 56 - 3 = 53 and 56 + 3 = 59 are prime.
a(14) = 265 is the first term that corresponds to more than one primitive Pythagorean triangle: sides (23, 264, 265) and (247, 96, 265).

Maple

N:= 10000: # for terms <= N
R:= NULL:
for m from 1 while m^2 < N do
  for n from 1 + (m mod 2) to m by 2 while m^2 + n^2 <= N do
    if igcd(m, n) > 1 then next fi;
    if isprime(2*m^2 + m*n + n^2) and isprime(2*m^2 + 3*m*n - n^2) then
  R:= R, m^2+n^2;
fi od od:
sort(convert({R}, list));

Crossrefs

Cf. A390886.

Sequence in context: A063588 A307286 A339955 * A117635 A268526 A018447

Adjacent sequences: A390246 A390247 A390248 * A390250 A390251 A390252

Keyword

nonn

Author

Will Gosnell and Robert Israel, Dec 01 2025

Status

approved