A393205 Primes p such that there is at least one Pythagorean triple with perimeter s and inradius r such that s + r^2 = p^2.

7, 17, 41, 97, 127, 239, 601, 937, 1471, 3079, 4201, 5081, 9127, 11369, 20231, 22367, 23977, 32801, 49727, 80191, 133121, 173729, 216743, 240319, 259201, 279047, 299881, 368551, 393577, 419711, 558863, 567841, 601127, 635689, 915367, 1610513, 1698751, 1742249, 1984639, 2060801, 2300927, 2769479

Offset

1,1

Links

Table of n, a(n) for n=1..42.

Robert Israel, Data for the first 58 terms

Example

a(3) = 41 is a term because 41 is prime and the Pythagorean triple (95, 168, 193) has perimeter 95 + 168 + 193 = 456 and inradius (95 + 168 - 193)/2 = 35 and 456 + 35^2 = 41^2.

Maple

N:= 10^14: # for terms <= sqrt(N)
Res:= NULL:
for x from 2 by 2 while 2*x^2 <= N do
  ym:= select(t -> t >= 1 and t <= x-1, [fsolve(2*x^2*yy - 6*x*yy^2 + 4*yy^3 + 2*x = 0)]);
  if ym = [] then ym:= x-1 else ym:= op(ym) fi;
  for y from 1 to ym by 2 do
    if igcd(x, y) > 1 then next fi;
    p:= 2*x*(x+y);
    r:= (x-y)*y;
    v:= p+r^2;
    if v > N then break fi;
    if issqr(v) and isprime(sqrt(v)) then Res:= Res, sqrt(v); fi
  od;
  for y from x-1 to ym by -2 do
        if igcd(x, y) > 1 then next fi;
    p:= 2*x*(x+y);
    r:= (x-y)*y;
    v:= p+r^2;
    if v > N then break fi;
    if issqr(v) and isprime(sqrt(v)) then Res:= Res, sqrt(v); fi
    fi
  od;
od:
sort([Res]);

Crossrefs

Cf. A390249, A392634, A390819, A390820, A392948, A392988.

Sequence in context: A193214 A184862 A194772 * A193546 A268255 A124965

Adjacent sequences: A393202 A393203 A393204 * A393206 A393207 A393208

Keyword

nonn

Author

Will Gosnell and Robert Israel, Feb 05 2026

Status

approved