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

Author

Wolfdieter Lang, Sep 11 2019

Keywords

• nonn
• easy

Comments

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).

Examples

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

References

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

Crossrefs

Cf. A002559, A005087, A324250, A327343.

Formula

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