A226693 Squarefree parts of A079896(n), n>= 1.

5, 2, 3, 13, 17, 5, 21, 6, 7, 29, 2, 33, 37, 10, 41, 11, 5, 3, 13, 53, 14, 57, 15, 61, 65, 17, 69, 2, 73, 19, 77, 5, 21, 85, 22, 89, 23, 93, 6, 97, 101, 26, 105, 3, 109, 7, 113, 29, 13, 30, 31, 5, 2, 129, 33, 133, 34, 137, 35, 141, 145, 37, 149, 38, 17, 39, 157, 10

Offset: 1

Author

Wolfdieter Lang, Jun 15 2013

Keywords

• nonn
• easy

Comments

a(n) is the squarefree part of the discriminant D(n) = A079896(n) of indefinite binary quadratic forms. Certain quadratic irrationals, called omega_p(D(n)), related to the principal indefinite form of discriminant D(n) are integers in the quadratic number field Q(sqrt(a(n))). See A226166 for the definition of these irrationals omega_p(D(n)) using the D. A. Buell reference, p. 31 and p. 26.

For discriminants D == 1 (mod 4) these squarefree parts are given in A226165. For D == 0 (mod 4) the squarefree parts are given in A002734 corresponding to A000037 = D/4.

References

• D. A. Buell, Binary Quadratic Forms, Springer, 1989.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..20001

Crossrefs

Cf. A079896, A226165, A002734.

Programs

Mathematica

SquareFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]); SquareFreePart /@ Select[ Range[160], ! IntegerQ[Sqrt[#]] && Mod[#, 4] < 2 &] (* Jean-François Alcover, Jun 25 2013 *)

PARI

A079896_list(N) = {
  my(n = 1, v = vector(N), top = 0);
  while (top < N, if (n%4 < 2 && !issquare(n), v[top++] = n); n++;);
  return(v);
};
apply(core, A079896_list(68)) \\ Gheorghe Coserea, Nov 10 2016

Formula

a(n) = squarefree part of D(n) = A079896(n), n >= 1, the numbers 0 and 1 (mod 4), not a square.

Extensions

Offset corrected by Robin Visser, Jun 01 2025