A089269 Squarefree numbers congruent to 1 or 2 mod 4.

1, 2, 5, 6, 10, 13, 14, 17, 21, 22, 26, 29, 30, 33, 34, 37, 38, 41, 42, 46, 53, 57, 58, 61, 62, 65, 66, 69, 70, 73, 74, 77, 78, 82, 85, 86, 89, 93, 94, 97, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 134, 137, 138, 141, 142, 145, 146, 149, 154, 157

Offset: 1

Author

Wolfdieter Lang, Nov 07 2003

Keywords

• nonn
• easy

Comments

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.

The asymptotic density of this sequence is 4/Pi^2 (A185199). - Amiram Eldar, Feb 23 2021

References

• Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 231-234.
• Arnold Scholz and Bruno Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Ch. 30.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Magma

[n: n  in [1..200] | IsSquarefree(n) and n mod 4 in [1,2]]; // Vincenzo Librandi, Oct 20 2017

Mathematica

Select[Range[200], MemberQ[{1, 2}, Mod[#, 4]]&& SquareFreeQ[#]&] (* Vincenzo Librandi, Oct 20 2017 *)

Extensions

Entry revised by N. J. A. Sloane, May 28 2014