A390819 Numbers k such that there is a primitive right triangle with perimeter - inradius = k^2.

8, 22, 23, 37, 43, 79, 106, 109, 137, 149, 172, 197, 211, 239, 254, 268, 284, 316, 319, 326, 352, 386, 401, 407, 428, 443, 463, 466, 473, 484, 487, 529, 548, 562, 569, 596, 617, 638, 641, 659, 667, 674, 688, 737, 746, 751, 764, 781, 788, 821, 823, 841, 842, 844, 851, 862, 869, 877, 904, 989, 991

Offset

1,1

Comments

Solutions to 2*m^2 + m*n + n^2 = k^2 where m and n are coprime and one of m and n is even.

Links

Robert Israel, Table of n, a(n) for n = 1..2500

Example

a(3) = 23 is a term because with m = 16 and n = 1, the primitive right triangle with sides a = m^2 - n^2 = 255, b = 2*m*n = 32 and c = m^2 + n^2 = 257 has perimeter a+b+c = 544, inradius (a+b-c)/2 = 15 and perimeter - inradius = 544 - 15 = 529 = 23^2.

Maple

N:= 1000: # for terms <= N
R:= {}:
for m from 1 while 2*m^2 < N^2 do
  for n from 1+(m mod 2) to m-1 by 2 do
    v:= 2*m^2 + m*n + n^2; if v > N^2 then break fi;
    if igcd(m, n) = 1 and issqr(v) then
      a:= m^2 - n^2; b:= 2*m*n; c:= m^2 + n^2; y:= sqrt(v);
      R:= R union {y}
    fi
od od:
sort(convert(R, list));

Crossrefs

Cf. A390820.

Sequence in context: A013630 A215600 A304301 * A236917 A064193 A301482

Adjacent sequences: A390816 A390817 A390818 * A390820 A390821 A390822

Keyword

nonn

Author

Will Gosnell and Robert Israel, Nov 20 2025

Status

approved