A327248 - OEIS

%I #21 Mar 02 2023 07:18:47

%S 6,210,1785,60639,915530,184030,14066106,80753867670,10973017315470,

%T 372759573255306,351745902037915,11949006236698685,86466986871277074,

%U 122261486084598,43869141307765893,35803482505852454889891,2162247909473892250092390,73452778286546376583337010

%N Squarefree part of A078522(n+1).

%C Also the squarefree part of (A001653(n+1)^2-1)/2 or of A002315(n)^2-1

%C Walsh shows that the system of simultaneous Pell equations x^2 - d*y^2 = z^2 - 2*d*y^2 = 1 has solutions in positive integers x, y, z if and only if d belongs to this sequence and, under the abc conjecture, this sequences grows exponentially.

%H P. G. Walsh, <a href="https://doi.org/10.4064/aa-82-1-69-76">On integer solutions to x^2 - dy^2 = 1, z^2 - 2dy^2 = 1</a>, Acta Arithmetica 82 (1997), 69-76.

%F a(n) = A007913(A078522(n+1)).

%e a(2) = 210 since A078522(3) = 840 = 210 * 2^2.

%o (PARI) a(n)={local(z=1+quadgen(8)); core(imag(z^(2*n+1))^2-1)}

%Y Cf. A007913, A001653, A002315, A078522.

%K nonn

%O 1,1

%A _Tomohiro Yamada_, Sep 15 2019

%E Missing a(11) inserted and more terms from _Georg Fischer_, Mar 02 2023