A379018 a(n) is the least prime p such that p^2 < n and n - p^2 is not squarefree, or -1 if there is no such p.

-1, -1, -1, -1, -1, -1, -1, 2, -1, -1, -1, 2, 2, -1, -1, 2, 3, 3, -1, 2, 3, 2, -1, 2, 3, -1, 3, 2, 2, -1, 2, 2, 3, 3, -1, 2, 3, -1, -1, 2, 3, -1, 5, 2, 3, -1, -1, 2, 2, 5, -1, 2, 2, 2, -1, 2, 3, 2, 3, 2, 3, -1, 3, 2, 3, -1, 2, 2, 3, 5, -1, 2, 3, 5, 5, 2, 3, -1, 2, 2, 3, -1, -1, 2, 2, -1, -1, 2, 3

Offset: 1

Author

Robert Israel, Dec 13 2024

Keywords

• sign

Comments

a(n) is the least p, if it exists, such that there is a solution to the equation n = p^2 + k * q^2 with p and q prime and k > 0.

Examples

			a(17) = 3 because 17 - 3^2 = 8 is not squarefree, while 17 - 2^2 = 13 is squarefree.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A013929, A378896.

Programs

Maple

f:= proc(n) local p;
  p:= 2;
  while p^2 <= n do
    if not numtheory:-issqrfree(n-p^2) then return p fi;
    p:= nextprime(p);
  od;
  -1
end proc:
map(f, [$1..200]);