A392988 Numbers k such that there is a primitive Pythagorean triple with perimeter k and inradius r such that p + r, p - r, p + r^2 and |p - r^2| are all prime.

12, 84, 132, 208, 260, 532, 552, 840, 864, 1012, 1276, 1628, 1820, 1892, 2340, 2444, 5548, 5720, 5900, 5928, 6136, 7668, 8832, 9372, 9900, 12672, 12948, 15048, 16132, 16940, 18540, 19180, 20592, 23088, 24384, 27180, 28380, 28724, 29868, 30988, 31824, 37332, 37408, 45576, 45880, 48488, 50388

Offset

1,1

Comments

The first term that corresponds to more than one value of the inradius is a(344) = 1915880, where the primitive Pythagorean triples (491208, 627655, 797017) and (627655, 491208, 797017) have perimeter 1915880 and inradii 160923 and 111867 respectively.

Links

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

Example

a(3) = 132 is a term because the primitive Pythagorean triple (11, 60, 61) has perimeter 132 and inradius 5, and 132 + 5 = 137, 132 - 5 = 127, 132 + 5^2 = 157 and 132 - 5^2 = 107 are all prime.

Maple

N:= 60000: # for terms <= N
S:= {}:
for x from 2 by 2 while 2*x^2<N do
  for y from 1 by 2 to min(x-1, (N-2*x^2)/(2*x)) do
       if igcd(x, y) > 1 then next fi;
       p:= 2*x^2 + 2*x*y;
       r:= x*y - y^2;
       if andmap(isprime, [p+r^2, abs(p-r^2), p+r, abs(p-r)]) then S:= S union {p} fi
od od:
sort(convert(S, list));

Crossrefs

Cf. A390249, A392948.

Sequence in context: A213347 A075476 A298977 * A213784 A085409 A384288

Adjacent sequences: A392985 A392986 A392987 * A392989 A392990 A392991

Keyword

nonn

Author

Will Gosnell and Robert Israel, Jan 29 2026

Status

approved