A226166 -id:A226166 - cp's OEIS frontend

A307372 One half of the row length of A324251: one half of the length of the reduced principal cycle for discriminant 4*D(n), with D(n) = A000037(n).

1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 2, 1, 1, 1, 3, 1, 3, 3, 2, 1, 1, 1, 2, 5, 1, 4, 2, 2, 2, 1, 1, 1, 1, 1, 3, 1, 5, 4, 3, 6, 2, 1, 1, 1, 3, 5, 3, 2, 1, 3, 7, 3, 2, 11, 2, 1, 1, 1, 5, 1

Offset: 1

Wolfdieter Lang, Apr 19 2019

nonn

This is a subsequence of A226166. See the formula.

For details on the cycles for the principal form F_p(n) = FR(n), the first reduced form of the not reduced Pell form F(n) = [1, 0, -D(n)], see A324251, also for references and a W. Lang link with Table 1, last column LCR(n) = 2*a(n).

Cf. A000037, A079896, A226166, A324251.

a(n) equals one half of the length of the reduced principal cycle of discriminant 4*D(n), with D(n) = A000037(n), for n >= 1.

a(n) = A226166(e(n)), where e(n) is the n-th even term of A079896, for n >= 1.

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

Original entry on oeis.org

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

Wolfdieter Lang, Jun 15 2013

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.

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

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

Cf. A079896, A226165, A002734.

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 *)

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

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

Offset corrected by Robin Visser, Jun 01 2025

A257161 The length of the period under Zagier-reduction of the principal indefinite quadratic binary form of discriminant D(n) = A079896(n).

Original entry on oeis.org

1, 2, 1, 3, 5, 4, 1, 2, 2, 5, 1, 4, 7, 6, 11, 3, 1, 2, 10, 7, 2, 7, 1, 11, 9, 8, 2, 4, 21, 7, 1, 2, 4, 9, 6, 21, 2, 3, 1, 27, 11, 10, 3, 5, 17, 6, 23, 16, 1, 2, 8, 11, 2, 15, 2, 6, 2, 27, 1

Offset: 1

Barry R. Smith, Apr 16 2015

nonn

A binary quadratic form A*x^2 + B*x*y + C*y^2 with integer coefficients A, B, and C and positive discriminant D = B^2 - 4*A*C is Zagier-reduced if A>0, C>0, and B>A+C. (This differs from the classical reduced forms defined by Lagrange.)  There are finitely many Zagier-reduced forms of given discriminant.
Zagier defines a reduction operation on binary quadratic forms with positive discriminants, which permutes the reduced forms.  The reduced forms are thereby partitioned into disjoint cycles.
There is a unique Zagier-reduced form with A=1 for each discriminant in A079896. The cycle containing this form is the principal cycle.  a(n) is the length of this cycle for the discriminant D=A079896(n).

			For n=4, the a(4) = 3 forms in the principal cycle of discriminant A079896(4) = 13 are x^2 + 5*x*y + 3*y^2, 3*x^2 + 5*x*y + y^2, and 3*x^2 + 7*x*y + 3*y^2.

D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.

Cf. A226166.

With D=n^2-4, a(n) equals the number of pairs (a,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), a > (sqrt(D) - k)/2, a exactly dividing (D-k^2)/4.

Offset corrected by Robin Visser, Jun 08 2025

cp's OEIS Frontend

A307372 One half of the row length of A324251: one half of the length of the reduced principal cycle for discriminant 4*D(n), with D(n) = A000037(n).

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

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A257161 The length of the period under Zagier-reduction of the principal indefinite quadratic binary form of discriminant D(n) = A079896(n).

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