A327248 Squarefree part of A078522(n+1).

6, 210, 1785, 60639, 915530, 184030, 14066106, 80753867670, 10973017315470, 372759573255306, 351745902037915, 11949006236698685, 86466986871277074, 122261486084598, 43869141307765893, 35803482505852454889891, 2162247909473892250092390, 73452778286546376583337010

Offset: 1

Author

Tomohiro Yamada, Sep 15 2019

Keywords

• nonn

Comments

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

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.

Examples

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

Links

• P. G. Walsh, On integer solutions to x^2 - dy^2 = 1, z^2 - 2dy^2 = 1, Acta Arithmetica 82 (1997), 69-76.

Crossrefs

Cf. A007913, A001653, A002315, A078522.

Programs

PARI

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

Formula

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

Extensions

Missing a(11) inserted and more terms from Georg Fischer, Mar 02 2023