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A386894 Markoff numbers that are powers of one odd prime or twice powers of one odd prime.

%I A386894 #23 Aug 13 2025 10:20:30
%S A386894 1,2,5,13,29,34,89,169,194,233,433,1597,2897,5741,7561,28657,33461,
%T A386894 43261,96557,426389,514229,646018,1686049,2012674,2922509,3276509,
%U A386894 11485154,21531778,94418953,253191266,321534781,433494437,780291637,1405695061,1475706146,2971215073,6684339842,19577194573
%N A386894 Markoff numbers that are powers of one odd prime or twice powers of one odd prime.
%C A386894 A subsequence of A000961 without numbers divisible by 4.
%C A386894 The powers of odd primes are given in A061345 (with offset 0).
%C A386894 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.
%D A386894 Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013.
%H A386894 Mong Lung Lang and Ser Peow Tan, <a href="https://arxiv.org/abs/math/0508443">A simple proof of the Markoff conjecture for prime powers</a>, arXiv:math/0508443 [math.NT], 2005.
%H A386894 Paul Schmutz, <a href="https://eudml.org/doc/165440">Systoles of arithmetic surfaces and the Markoff spectrum</a>, Math. Ann. 305 (1996), no. 1, 191-203.
%H A386894 Ying Zhang, <a href="https://arxiv.org/abs/math/0606283">An elementary proof of uniqueness of Markoff numbers which are prime powers</a>, arXiv:math/0606283 [math.NT], 2007.
%F A386894 Markoff numbers of the form 2^j*p^k, with an odd prime p, j = 0 or 1, and k >= 0, ordered strictly increasing.
%e A386894 26 = 2*13 is not a Markoff number, hence not in the present sequence.
%e A386894 610 = 2*5*61 is a Markoff number but not a prime power nor is 305 a prime power.
%t A386894 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 *)
%o A386894 (SageMath)
%o A386894 def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:
%o A386894     # Using function 'MarkovNumbers' from A002559.
%o A386894     M = MarkovNumbers(len, MAX)
%o A386894     U = set([1])
%o A386894     for m in M:  # if m is a Markov number and ...
%o A386894         z = ZZ(m)
%o A386894         if is_prime_power(z) or (is_even(z) and is_prime_power(z//2)):
%o A386894             U.add(m)
%o A386894     return sorted(U)
%o A386894 # Balance required sequence length and search depth.
%o A386894 print(A386894List(len=120, MAX=10**12))  # _Peter Luschny_, Aug 12 2025
%Y A386894 Cf. A000040, A000961, A002559, A061345, A256395.
%K A386894 nonn,easy
%O A386894 1,2
%A A386894 _Wolfdieter Lang_, Aug 07 2025