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A307530 Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.

%I A307530 #12 Apr 13 2019 07:11:50
%S A307530 3,23,47,59,61,79,97,137,139,163,167,191,193,223,251,281,283,313,317,
%T A307530 349,353,359,389,397,431,433,439,479,521,523,563,569,571,613,617,619,
%U A307530 659,661,673,719,727,769,773,823,827,829,839,881,883,887,941,947,953,1009
%N A307530 Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.
%C A307530 Misak and Ulas prove that the density of primes with k ones is 1/(Fibonacci(k+3)*Fibonacci(k+1)) = 1/3, here with k=1 (a single 1).
%H A307530 Piotr Miska, Maciej Ulas, <a href="https://arxiv.org/abs/1904.03404">On consecutive 1's in continued fractions expansions of square roots of prime numbers</a>, arXiv:1904.03404 [math.NT], 2019. See Corollary 4.3. p. 13.
%H A307530 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%e A307530 For p = 3,  we have [1; 1, 2, ...]; see A040001.
%e A307530 For p = 27, we have [4; 1, 3, ...]; see A010127.
%e A307530 For p = 47, we have [6; 1, 5, ...]; see A010137.
%o A307530 (PARI) isok(p) = my(cf = contfrac(sqrt(p))); (cf[2] == 1) && (cf[3] != 1);
%o A307530 lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", ")));
%Y A307530 Cf. A040001, A010127, A010137, A307453.
%K A307530 nonn
%O A307530 1,1
%A A307530 _Michel Marcus_, Apr 13 2019