A325421 Numbers k coprime to 10 such that there are exactly two values of A for which k^2+4*A and k^2-4*A are perfect squares.

169, 289, 507, 841, 867, 1183, 1369, 1521, 1681, 1859, 2023, 2523, 2601, 2809, 3179, 3211, 3549, 3721, 3887, 4107, 4563, 5043, 5239, 5329, 5491, 5577, 5887, 6069, 6647, 7267, 7569, 7803, 7921, 7943, 8281, 8427, 8959, 9251, 9409, 9537, 9583, 9633, 9971

Offset

1,1

Comments

These are the odd integers k, not a multiple of 5, such that k^2 is an arithmetic mean of two other odd perfect squares in exactly two ways.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

169 is a term since 169^2+-4*(5070) and 169^2+-4*(7140) are all perfect squares.

Maple

filter:= proc(k) local A, count, y, v;
  if k mod 5 = 0 then return false fi;
  count:= 0:
  for y from k+2 by 2 do
    v:= 2*k^2-y^2;
    if v < 0 then break fi;
    if issqr(v) then count:= count+1 fi;
  od;
  count = 2
end proc:
select(filter, [seq(i, i=1..10000, 2)]); # Robert Israel, Sep 27 2025

Prog

(PARI) ok(k)={if(k%2==0||k%5==0, 0, my(k2=k^2, L=List()); forstep(i=1, k-1, 2, my(d=k2-i^2); if(issquare(k2+d), listput(L, i))); #L==2)}
for(k=1, 10000, if(ok(k), print1(k, ", "))) \\ Andrew Howroyd, Sep 06 2019

Crossrefs

Cf. A002144, A309812.

Sequence in context: A383691 A020249 A287391 * A296304 A156159 A099011

Adjacent sequences: A325418 A325419 A325420 * A325422 A325423 A325424

Keyword

nonn

Author

Mohsin A. Shaikh, Sep 06 2019

Extensions

a(28)-a(43) from Andrew Howroyd, Sep 06 2019

Status

approved