A383898 a(n) is the smallest nonnegative integer k such that n + k and n*k are squares, or -1 if there is no such number.

0, 2, -1, 0, 20, -1, -1, 8, 0, 90, -1, -1, 4212, -1, -1, 0, 272, 18, -1, 5, -1, -1, -1, -1, 0, 650, -1, -1, 142100, -1, -1, 32, -1, -1, -1, 0, 1332, -1, -1, 360, 41984, -1, -1, -1, 180, -1, -1, -1, 0, 50, -1, 117, 1755572, -1, -1, -1, -1, 568458, -1, -1, 53872730964

Offset: 1

Author

Gonzalo Martínez, May 15 2025

Keywords

• sign

Comments

a(m^2) = 0, for all positive integers m.

If m is not in A245226 then a(m) = -1. Indeed, if (u, v) is the smallest solution of the equation x^2 - m*y^2 = m, then m + m*v^2 = u^2 and m*(m*v^2) = (m*v)^2. Therefore, a(m) = m*v^2.

The sequence A383734 contains the numbers k such that 2 + k and 2*k are squares, where A383734(2) = a(2). Similarly, A382209(1) = a(10).

a(181) <= 223502910856088814900 = 181*1111225770^2. - Michel Marcus, May 24 2025

Examples

			If n = 5, then 20 satisfies that 5 + 20 = 5^2 and 5*20 = 10^2, where 20 is the smallest integer greater than 5 with this property. So, a(5) = 20.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..180

Crossrefs

Cf. A382209, A383734, A245226.

Programs

PARI

isok(m) = if (issquare(4*m), 1, #qfbsolve(Qfb(1, 0, -m), m, 2)); \\ A245226
a(n) = if (!isok(n), return(-1)); my(k=0); while (!(issquare(n+k) && issquare(n*k)), k++); k; \\ Michel Marcus, May 21 2025

PARI

isok(m) = if (issquare(4*m), 1, #qfbsolve(Qfb(1, 0, -m), m, 2)); \\ A245226
a(n) = if (!isok(n), return(-1)); my(x = sqrtint(n)); while (! issquare(n*(x^2 - n)), x++); x^2-n; \\ Michel Marcus, May 24 2025