A323728 a(n) is the smallest number k such that both k-2*n and k+2*n are squares.

2, 5, 10, 8, 26, 13, 50, 20, 18, 29, 122, 25, 170, 53, 34, 32, 290, 45, 362, 41, 58, 125, 530, 52, 50, 173, 90, 65, 842, 61, 962, 80, 130, 293, 74, 72, 1370, 365, 178, 89, 1682, 85, 1850, 137, 106, 533, 2210, 100, 98, 125, 298, 185, 2810, 117, 146, 113, 370

Offset: 1

Author

Daniel Suteu, Jan 25 2019

Keywords

• nonn

Comments

When n is a prime number, a(n) is greater than all the previous terms.

If n = 4*x*y, then a(n) is the smallest integer solution of the form 4*(x^2 + y^2), with rational values x and y.

Examples

			For n = 3, a(3) = 10, which is the smallest integer k such that k+2*n and k-2*n are both squares: 10+2*3 = 4^2 and 10-2*3 = 2^2.
For n=1..10, the following {a(n)-2*n, a(n)+2*n} pairs of squares are produced: {0, 4}, {1, 9}, {4, 16}, {0, 16}, {16, 36}, {1, 25}, {36, 64}, {4, 36}, {0, 36}, {9, 49}.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Difference of two squares

Crossrefs

Cf. A000290, A029744, A033676. A033677, A056737, A061057, A063655, A087711.

Programs

Maple

f:= proc(n) local d;
d:= max(select(t -> t^2 <= n, numtheory:-divisors(n)));
d^2 + (n/d)^2
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

Mathematica

Array[Block[{k = 1}, While[Nand @@ Map[IntegerQ, Sqrt[k + 2 {-#, #}]], k++]; k] &, 57] (* Michael De Vlieger, Feb 17 2019 *)

PARI

a(n) = for(k=2*n, oo, if(issquare(k+2*n) && issquare(k-2*n), return(k)));

PARI

a(n) = my(d=divisors(n)); vecmin(vector(#d, k, 4*((d[k]/2)^2 + (n/d[k]/2)^2)));

Formula

a(n^2) = 2 * n^2.

a(p) = p^2 + 1, for p prime.

a(n) = A063655(n)^2 - 2*n.

a(n) = A056737(n)^2 + 2*n.

a(n!) = A061057(n)^2 + 2*n!.

a(n) = A033676(n)^2 + A033677(n)^2. - Robert Israel, Feb 17 2019

a(n) = Min_{d|n} ((n/d)^2 + d^2). - Ridouane Oudra, Mar 17 2024