A378896 Numbers k such that k - p^2 is squarefree for every prime p < sqrt(k).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 19, 23, 26, 30, 35, 38, 39, 42, 46, 47, 51, 55, 62, 66, 71, 78, 82, 83, 86, 87, 91, 95, 110, 111, 114, 118, 119, 122, 127, 131, 138, 143, 155, 158, 163, 167, 182, 183, 186, 190, 191, 195, 203, 206, 210, 215, 222, 226, 227, 230, 231, 235, 239, 255, 258, 262

Offset: 1

Author

Robert Israel, Dec 14 2024

Keywords

• nonn

Comments

Numbers k such that there is no solution to k = p^2 + m * q^2 with p and q prime and m > 0.

Numbers k such that A379018(k) = -1.

Examples

			a(10) = 11 is a term because both 11 - 2^2 = 7 and 11 - 3^2 = 2 are squarefree, while 11 - 5^2 < 0.
		

Links

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

Crossrefs

Cf. A005117, A379018, A331802.

Programs

Maple

filter:= proc(n) local p;
  p:= 2;
  while p^2 <= n do
    if not numtheory:-issqrfree(n-p^2) then return false fi;
    p:= nextprime(p);
  od;
  true
end proc:
select(filter, [$1..300]);

Mathematica

sfpQ[n_]:=With[{prs=Select[Prime[Range[PrimePi[Sqrt[n]]]],#Harvey P. Dale, Mar 24 2025 *)