A324250 -id:A324250 - cp's OEIS frontend

A327342 a(n) gives the number of distinct odd prime divisors of m(n) = A002559(n) (Markoff numbers).

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 3, 1, 1, 2, 3, 2, 2, 2, 3, 2, 1, 3, 3, 2, 3, 1, 3, 3, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 1, 3, 2, 1, 4, 1, 2, 2, 4, 2, 1, 3, 3, 4, 3, 1, 1, 2, 2, 2, 1, 3, 2, 3, 2

Offset: 1

Wolfdieter Lang, Sep 11 2019

These sequence members appear as exponents of 2 in the number of representative parallel primitive forms for binary quadratic forms of discriminant Disc(n) = 9*m(n)^2 - 4 and representation of -m(n)^2. The reduced (primitive) principal form of this discriminant is F_p(n; X, Y) = X^2 + b(n)*X*Y - b(n)*Y^2, written also as F_p(n) = [1, b(n), -b(n)], with b(n) = 3*m(n) - 2 = A324250(n). This form representing -m(n)^2 is important for the determination of Markoff triples MT(n).

For more details see A327343(n) = 2^a(n). The Frobenius-Markoff uniqueness conjecture on ordered triples with largest member m(n) is certainly true for m(n) if a(n) = 0 (so-called singular cases) or 1. See the Aigner reference, p. 59, Corollary 3.20, for n >= 3 (the a(n) = 1 cases).

			For the examples a(6) = 1 and a(12) = 2 see A327343.

Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013.

Cf. A002559, A005087, A324250, A327343.

a(n) = number of distinct odd prime divisors of m(n) = A002559(n), for n >= 1.

a(n) = A005087(A002559(n)). - Michel Marcus, Sep 18 2023

A327343 a(n) gives the number of representative parallel primitive forms for binary quadratic forms of discriminant Disc(n) = 9*m(n)^2 - 4 and representation of -m(n)^2, with m(n) = A002559(n) (Markoff numbers).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 8, 8, 4, 2, 4, 2, 2, 4, 4, 4, 4, 8, 2, 2, 8, 2, 2, 4, 8, 4, 4, 4, 8, 4, 2, 8, 8, 4, 8, 2, 8, 8, 4, 4, 4, 8, 2, 4, 4, 4, 4, 4, 4, 2, 8, 4, 2, 16, 2, 4, 4, 16, 4, 2, 8, 8, 16, 8, 2, 2, 4, 4, 4, 2, 8, 4, 8, 4

Offset: 1

Wolfdieter Lang, Sep 13 2019

For the definition of parallel forms for an indefinite binary quadratic form with discriminant Disc and representation of an integer k see, e.g., the Buell, Scholz-Schoeneberg references or the W. Lang link, section 3, with a scanning prescription.
For the Markoff case Disc(n) = 9*m(n) - 4 = b(n)*(b(n)+2), with m = A002559 and b = A324250.
The Markoff form MF(n;x,y) = x^2 - 3*m(n)*x*y + y^2, also written as MF(n) = [1, -3*m(n), 1], representing -m(n)^2, has as first reduced form the principal form F_p(n;X,Y) = X^2 + b(n)*X*Y - b(n)*Y^2, or F_p(n) = [1, b(n), -b(n)], where the connection is X = x-y, Y = x, or x = Y, y = Y - X. Hence X <= 0 for x <= y.
Only proper solutions (gcd(X, Y) = 1) are of interest. Also only primitive representative parallel forms FPa(n;i), for  i = 1, 2, ..., #FPa(n), are considered.
In the present case it is possible to give directly the prescription for the primitive representative parallel forms (rpapfs). This is done for the even m(n) == 2 (mod 32) case and the odd case m(n) == 1 (mod 4) separately.
These rpapfs are written as FPa(n;i) = [-m(n)^2, B(n,i), -C(n,i)]. Their number a(n) =  #FPa(n) can be found from congruences with an application of the Chinese remainder theorem and the lifting theorem (see Apostol, Theorem 5.26, pp. 118-119, and Theorem 5.30, pp. 121-122 (only part (a) is effective here)). The existence of two solutions for each odd prime modulus is important as input for the lifting to higher prime powers. For each of the singular cases m(1) = 1 and m(2) = 2, without odd prime divisors, there is only one rpapf.
The Frobenius-Markoff uniqueness conjecture is certainly true for m(n) if a(n) = 1 or a(n) = 2. In the latter case the two rpapfs have to be equivalent to the principal form F_p(n), because the known solution implied by the ordered triple MT(n) = (x(n), y(n), m(n)) has an unordered partner solution which after ordering becomes (x(n), y'(n), m(n')) with y'(n) = m(n) and m(n') = 3*x(n)*m(n) - y(n) >= m(n).
See A327344 for details on the congruences which determine the rpapfs.

			n = 6: m(6) = 34 = 2*17, a(6) = 2. The (primitive) reduced principal form is F_p(6) = [1, 100, -100], and both representative parallel primitive forms are connected to this form via an equivalence transformation. The two proper fundamental solutions with X < 0 of F_p(6) = -34^2 are (X, Y)_1 = (-12, 1) and (X, Y)_2 = (-88, 1). They belong to the ordered Markoff triple MT(6) = (1, 13, 34) and the unordered one (1, 89, 34), respectively. The latter triple has 89 = 3*1*34 - 13, and is the ordered triple (1, 34, 89), not of interest in the search for ordered solution with maximum m(6).
Note that there are other proper fundamental positive solutions coming from the imprimitive form F = [4, 96, -74], namely (X, Y)_3 = (19, 26) and (X, Y)_4 = (133, 178) which are not counted here.
n = 12: m(12) = 610 = 2*5*61, a(12) = 4. The reduced principal form F_p(12) = [1, 1828, -1828], representing -610^2, has only two proper fundamental solutions with X < 0, Y > 0: (X, Y)_1 = (-232, 1), corresponding to the ordered Markoff triple MT(12) = (1, 233, 610), and (X, Y)_2 = (-1596, 1), corresponding to the unordered triple (1, 1597, 610). These solutions follow from the rpapfs [-372100, 742836, -370735] with t-tuple (-1, 231) and [-372100, 1364, 1] with t-tuple (1596), respectively.  The other two such proper fundamental solutions are (X, Y)_3 = (-6, 25) for the reduced form F(12) = [625, 1664, -232], and (X, Y)_4 = (-25, 6) for the associated form Fbar(12) = [-232, 1664, 625], both  representing -m(12)^2. These last two reduced forms belong to different (associated) 8-cycles. The corresponding rpapfs are [-372100, 623764, -261407] and [-372100, 120436, -9743].

Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013.
Tom M. Apostol, Introduction to Analytic Number Theory, 1976, Springer.
D. A. Buell, Binary quadratic forms, 1989, Springer, p. 49 (f').
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 105, eq. 129.

Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs, section 3.

Cf. A002559, A324250, A327342, A327344.

a(n) = 2^A327342(n), n >= 1, where A327342(n) is the number of distinct odd primes dividing m(n).

a(n) = number of representative parallel primitive forms (rpapfs) for discriminant Disc(n) = 9*m(n)^2 - 4 = b(n)*(b(n) + 4), with m(n) = A002559(n) and b(n) = A324250(n).

A308687 a(n) = A305312(n)/4 if A305312(n)is even and a(n) = (A305312(n) - 1)/4 if A305312(n) is odd, for n >= 1.

Original entry on oeis.org

1, 8, 55, 379, 1891, 2600, 17821, 64261, 84680, 122149, 421849, 837224, 2183005, 3950155, 5738419, 18883369, 39331711, 74157931, 94070600, 128629621, 185381839, 269583560, 486268651, 1847753209, 2519186671, 3192137000, 4210906771, 6000283981, 8707689224, 12664688905, 20977322059, 41089519729, 85578188905, 86805069128, 195388310755, 409067053471

Offset: 1

Wolfdieter Lang, Jul 15 2019

nonn

These numbers a(n), depending on the parity of the discriminants of Markoff forms Disc(n) = b(n)*(b(n) + 4) = A305312(n), with b(n) = A324250(n), enter the definition of representative parallel forms of Disc(n) and representation -m(n)^2, where m(n) = A002559(n) = (b(n) + 2)/3 are the Markoff numbers, in the following way. FPara(n) := [-m(n)^2, 2*j(n), -(j^2(n) - a(n))/m(n)^2] or [-m(n)^2, 2*j(n) + 1, -(j(n)^2 +j(n) - a(n))/m(n)^2], if Disc(n) is even or odd, respectively, with j(n) from the interval [0, m(n)^2 - 1] such that the third member of FPara(n) becomes an integer. See the W. Lang link in A324251, section 3 for representative parallel forms, and the Buell and Scholz-Schoeneberg references given there.

The trivial solution (x = 1, y = 0) of each of the #rpapfs (number of representative parallel and primitive forms) Fpara(n;k), for k = 1, 2, ..., #rpapfs, representing -m(n)^2 leads to a fundamental solution of any primitive form F = [a, b, c] = a*x^2 + b*x*y + c*y^2 of discriminant Disc := b^2 - 4*a*c and representing - m(n)^2, by a certain proper (determinant +1) equivalence transformation. For the Markoff triples the principal reduced form F_p = [1, b(n), -b(n)], representing -m(n)^2 is of interest. It is a member of a 2-cycle of reduced forms together with F = [-b(n), b(n), 1].

Cf. A002559, A305312, A324250.

a(n) = A305312(n)/4 or a(n) = (A305312(n) - 1)/4 if A305312(n) is even or odd, respectively, where A305312(n) = Disc(n) = b(n)*(b(n) + 4) with b(n) = 3*m(n) - 2 = A324250(n), and m(n) = A002559(n), for n >= 1.

cp's OEIS Frontend

A327342 a(n) gives the number of distinct odd prime divisors of m(n) = A002559(n) (Markoff numbers).

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A327343 a(n) gives the number of representative parallel primitive forms for binary quadratic forms of discriminant Disc(n) = 9*m(n)^2 - 4 and representation of -m(n)^2, with m(n) = A002559(n) (Markoff numbers).

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A308687 a(n) = A305312(n)/4 if A305312(n)is even and a(n) = (A305312(n) - 1)/4 if A305312(n) is odd, for n >= 1.

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