A386894 - OEIS

A386894

Markoff numbers that are powers of one odd prime or twice powers of one odd prime.

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 646018, 1686049, 2012674, 2922509, 3276509, 11485154, 21531778, 94418953, 253191266, 321534781, 433494437, 780291637, 1405695061, 1475706146, 2971215073, 6684339842, 19577194573

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OFFSET

1,2

COMMENTS

A subsequence of A000961 without numbers divisible by 4.

The powers of odd primes are given in A061345 (with offset 0).

These Markoff numbers (see A002559) have been proved to obey the Frobenius-Markoff uniqueness conjecture. See Aigner, Corollary 3.20, p. 59, and there the references [4] A. Baragar, [18] J. O. Button, and [119] Ying Zhang.

REFERENCES

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013.

LINKS

Table of n, a(n) for n=1..38.

Mong Lung Lang and Ser Peow Tan, A simple proof of the Markoff conjecture for prime powers, arXiv:math/0508443 [math.NT], 2005.

Paul Schmutz, Systoles of arithmetic surfaces and the Markoff spectrum, Math. Ann. 305 (1996), no. 1, 191-203.

Ying Zhang, An elementary proof of uniqueness of Markoff numbers which are prime powers, arXiv:math/0606283 [math.NT], 2007.

FORMULA

Markoff numbers of the form 2^j*p^k, with an odd prime p, j = 0 or 1, and k >= 0, ordered strictly increasing.

EXAMPLE

26 = 2*13 is not a Markoff number, hence not in the present sequence.

610 = 2*5*61 is a Markoff number but not a prime power nor is 305 a prime power.

MATHEMATICA

MAX=10^11; data=NestWhile[Select[Union[Sort/@Flatten[Table[{a, b, 3a b -c}/.MapThread[Rule, {{a, b, c}, #}]&/@Map[RotateLeft[ii, #]&, Range[3]], {ii, #}], 1]], Max[#]<MAX&]&, {{1, 1, 1}, {1, 1, 2}}, UnsameQ, 2]; m=data//Flatten//Union; Select[m, PrimeNu@#<2||PrimeNu[#/2]<2&] (* James C. McMahon, Aug 12 2025 *)

PROG

(SageMath)

def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:

# Using function 'MarkovNumbers' from A002559.

M = MarkovNumbers(len, MAX)

U = set([1])

for m in M: # if m is a Markov number and ...

z = ZZ(m)

if is_prime_power(z) or (is_even(z) and is_prime_power(z//2)):

U.add(m)

return sorted(U)

# Balance required sequence length and search depth.