A039957 Squarefree numbers congruent to 3 mod 4.

3, 7, 11, 15, 19, 23, 31, 35, 39, 43, 47, 51, 55, 59, 67, 71, 79, 83, 87, 91, 95, 103, 107, 111, 115, 119, 123, 127, 131, 139, 143, 151, 155, 159, 163, 167, 179, 183, 187, 191, 195, 199, 203, 211, 215, 219, 223, 227, 231, 235, 239, 247, 251, 255

Offset: 1

Author

R. K. Guy

Keywords

• nonn
• easy
• nice

Comments

Negatives of odd fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 224-230. See 4*A089269 = A191483 for the even case and A003657 for combined even and odd numbers. - Wolfdieter Lang, Nov 07 2003

The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

References

• Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
• Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• A. M. Legendre, Diviseurs de la forme t^2+au^2 a étant un nombre de la forme 4n-1, Essai sur la Théorie des Nombres An VI, Table V.

Crossrefs

Cf. A039955, A039956, A185197, A191483.

Programs

Haskell

a039957 n = a039957_list !! (n-1)
a039957_list = filter ((== 3) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

Magma

[4*n+3: n in [0..63] | IsSquarefree(4*n+3)];  // Bruno Berselli, Mar 04 2011

Mathematica

fQ[n_] := SquareFreeQ[n] && Mod[n, 4] == 3; Select[ Range@ 258, fQ] (* Robert G. Wilson v, Mar 02 2011 *)
Select[Range[3,300,4],SquareFreeQ] (* Harvey P. Dale, Mar 08 2015 *)

PARI

is(n)=n%4==3 && issquarefree(n) \\ Charles R Greathouse IV, Feb 07 2017

Extensions

Offset corrected