A387673 a(n) is the least number that is the area of a primitive Pythagorean triangle and is the sum of 2*n consecutive primes.

30, 60, 180, 180, 1320, 990, 1710, 23760, 1320, 1560, 30600, 13566, 4290, 1710, 567630, 391986, 2574, 51330, 9690, 46620, 4620, 18096, 249690, 32130, 9690, 249690, 106260, 262080, 2734374, 7980, 1139880, 328950, 666666, 17550, 215940, 1144920, 100464, 355680, 45144, 2477574, 2702700, 82110

Offset

1,1

Comments

Can the area of a primitive Pythagorean triangle can be the sum of an odd number of consecutive primes? The area is always a multiple of 6, so this would require the first of the primes to be 2.

Links

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

Example

a(4) = 180 because the Pythagorean triangle with sides 9, 40, 41 has area 180, and 180 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 is the sum of 8 consecutive primes, and no smaller number works.

Maple

N:= 10^6: # for terms involving sums of primes <= N
isA024365:= proc(a) local eq, xx, yy;
  eq:= (x^2-y^2)*x*y - a;
  for xx in numtheory:-divisors(a) do
    for yy in map(t -> rhs(op(t)), [isolve(subs(x=xx, eq))]) do
      if igcd(xx, yy) = 1 and xx+yy mod 2 = 1 then return true fi
  od od;
  false
end proc:
P:= select(isprime, [2, seq(i, i=3..10^6, 2)]):
PS:= ListTools:-PartialSums([0, op(P)]): nPS:= nops(PS):
g:= proc(n)
  local i, v;
  for i from 1 to nPS-2*n do
    v:= PS[i+2*n]-PS[i];
    if isA024365(v) then return v fi
  od;
  FAIL
end proc:
R:= NULL:
for n from 1 do
v:= g(n);
if v = FAIL then break fi;
R:= R, v;
od:
R;

Crossrefs

Cf. A024365, A050936, A383395.

Sequence in context: A108454 A235483 A386252 * A064783 A228877 A386517

Adjacent sequences: A387670 A387671 A387672 * A387674 A387675 A387676

Keyword

nonn

Author

Will Gosnell and Robert Israel, Nov 30 2025

Status

approved