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Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.
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%I #12 Apr 13 2019 07:11:50

%S 3,23,47,59,61,79,97,137,139,163,167,191,193,223,251,281,283,313,317,

%T 349,353,359,389,397,431,433,439,479,521,523,563,569,571,613,617,619,

%U 659,661,673,719,727,769,773,823,827,829,839,881,883,887,941,947,953,1009

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

%C 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 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 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%e For p = 3, we have [1; 1, 2, ...]; see A040001.

%e For p = 27, we have [4; 1, 3, ...]; see A010127.

%e For p = 47, we have [6; 1, 5, ...]; see A010137.

%o (PARI) isok(p) = my(cf = contfrac(sqrt(p))); (cf[2] == 1) && (cf[3] != 1);

%o lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", ")));

%Y Cf. A040001, A010127, A010137, A307453.

%K nonn

%O 1,1

%A _Michel Marcus_, Apr 13 2019