A325421 - OEIS

%I #55 Sep 28 2025 15:44:34

%S 169,289,507,841,867,1183,1369,1521,1681,1859,2023,2523,2601,2809,

%T 3179,3211,3549,3721,3887,4107,4563,5043,5239,5329,5491,5577,5887,

%U 6069,6647,7267,7569,7803,7921,7943,8281,8427,8959,9251,9409,9537,9583,9633,9971

%N 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.

%C 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.

%H Robert Israel, <a href="/A325421/b325421.txt">Table of n, a(n) for n = 1..1000</a>

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

%p filter:= proc(k) local A,count,y,v;

%p   if k mod 5 = 0 then return false fi;

%p   count:= 0:

%p   for y from k+2 by 2 do

%p     v:= 2*k^2-y^2;

%p     if v < 0 then break fi;

%p     if issqr(v) then count:= count+1 fi;

%p   od;

%p   count = 2

%p end proc:

%p select(filter, [seq(i,i=1..10000,2)]); # _Robert Israel_, Sep 27 2025

%o (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)}

%o for(k=1, 10000, if(ok(k), print1(k, ", "))) \\ _Andrew Howroyd_, Sep 06 2019

%Y Cf. A002144, A309812.

%K nonn

%O 1,1

%A _Mohsin A. Shaikh_, Sep 06 2019

%E a(28)-a(43) from _Andrew Howroyd_, Sep 06 2019