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

%I A378896 #41 Mar 24 2025 11:57:41
%S A378896 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,
%T A378896 66,71,78,82,83,86,87,91,95,110,111,114,118,119,122,127,131,138,143,
%U A378896 155,158,163,167,182,183,186,190,191,195,203,206,210,215,222,226,227,230,231,235,239,255,258,262
%N A378896 Numbers k such that k - p^2 is squarefree for every prime p < sqrt(k).
%C A378896 Numbers k such that there is no solution to k = p^2 + m * q^2 with p and q prime and m > 0.
%C A378896 Numbers k such that A379018(k) = -1.
%H A378896 Robert Israel, <a href="/A378896/b378896.txt">Table of n, a(n) for n = 1..10000</a>
%e A378896 a(10) = 11 is a term because both 11 - 2^2 = 7 and 11 - 3^2 = 2 are squarefree, while 11 - 5^2 < 0.
%p A378896 filter:= proc(n) local p;
%p A378896   p:= 2;
%p A378896   while p^2 <= n do
%p A378896     if not numtheory:-issqrfree(n-p^2) then return false fi;
%p A378896     p:= nextprime(p);
%p A378896   od;
%p A378896   true
%p A378896 end proc:
%p A378896 select(filter, [$1..300]);
%t A378896 sfpQ[n_]:=With[{prs=Select[Prime[Range[PrimePi[Sqrt[n]]]],#<Sqrt[n]&]},AllTrue[n-prs^2,SquareFreeQ]]; Select[Range[300],sfpQ] (* _Harvey P. Dale_, Mar 24 2025 *)
%Y A378896 Cf.  A005117, A379018, A331802.
%K A378896 nonn
%O A378896 1,2
%A A378896 _Robert Israel_, Dec 14 2024