login
Squarefree numbers congruent to 1 or 2 mod 4.
5

%I #24 Sep 08 2022 08:45:12

%S 1,2,5,6,10,13,14,17,21,22,26,29,30,33,34,37,38,41,42,46,53,57,58,61,

%T 62,65,66,69,70,73,74,77,78,82,85,86,89,93,94,97,101,102,105,106,109,

%U 110,113,114,118,122,129,130,133,134,137,138,141,142,145,146,149,154,157

%N Squarefree numbers congruent to 1 or 2 mod 4.

%C a(n) = one-fourth of the (negated) fundamental even discriminants D := b^2-4*a*c<0 of positive definite binary quadratic forms F=a*x^2+b*x*y+c*y^2. See A039957 for the odd numbers and A003657 for the combined even and odd numbers.

%C The asymptotic density of this sequence is 4/Pi^2 (A185199). - _Amiram Eldar_, Feb 23 2021

%D Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 231-234.

%D Arnold Scholz and Bruno Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Ch. 30.

%H N. J. A. Sloane, <a href="/A089269/b089269.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Range[200], MemberQ[{1, 2}, Mod[#, 4]]&& SquareFreeQ[#]&] (* _Vincenzo Librandi_, Oct 20 2017 *)

%o (Magma) [n: n in [1..200] | IsSquarefree(n) and n mod 4 in [1,2]]; // _Vincenzo Librandi_, Oct 20 2017

%Y Cf. A000003, A005117, A039957, A003657, A185199.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Nov 07 2003

%E Entry revised by _N. J. A. Sloane_, May 28 2014