|
%I #22 Mar 17 2023 14:30:11
%S 33,39,85,119,133,253,377,403,527,629,703,943,989,1363,1537,1643,2183,
%T 2257,2747,2881,3053,3139,3337,3431,4187,4399,4897,5251,5429,6499,
%U 6767,6887,6901,7171,7313,7373,7519,7597,7811,7957,8453,8611,8927,9379,11303
%N Squarefree semiprimes k such that (m+1)^2-k is also a square, where m = ceiling(sqrt(k)).
%C Original name: Squarefree semiprimes k such that the second-next perfect square minus k is a perfect square.
%e 3*11 = 33, 49-33 = 16 -> 4, 7-4 = 3, 7+4 = 11;
%e 3*13 = 39, 64-39 = 25 -> 5, 8-5 = 3, 8+5 = 13.
%t f1[n_] := Last/@FactorInteger[n] == {1,1}; f2[n_] := IntegerQ[Sqrt[(Ceiling[Sqrt[n]] + 1)^2 - n]]; lst={}; Do[If[f1[n] && f2[n], AppendTo[lst,n]], {n,8!}]; lst
%t Select[Range[12000],PrimeOmega[#]==2&&SquareFreeQ[#]&&IntegerQ[Sqrt[ (Ceiling[ Sqrt[#]]+1)^2-#]]&] (* _Harvey P. Dale_, Mar 17 2023 *)
%o (PARI) isok(k) = issquarefree(k) && (bigomega(k)==2) && issquare((ceil(sqrt(k))+1)^2-k); \\ _Michel Marcus_, Nov 27 2019
%Y Cf. A006881, A180655.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Sep 15 2010
%E Original name replaced (using an Apr 19 2012 Comments entry from _M. F. Hasler_) by _Jon E. Schoenfield_, Nov 25 2019
|
