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

3, 23, 47, 59, 61, 79, 97, 137, 139, 163, 167, 191, 193, 223, 251, 281, 283, 313, 317, 349, 353, 359, 389, 397, 431, 433, 439, 479, 521, 523, 563, 569, 571, 613, 617, 619, 659, 661, 673, 719, 727, 769, 773, 823, 827, 829, 839, 881, 883, 887, 941, 947, 953, 1009

Offset: 1

Author

Michel Marcus, Apr 13 2019

Keywords

• nonn

Comments

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).

Examples

			For p = 3,  we have [1; 1, 2, ...]; see A040001.
For p = 27, we have [4; 1, 3, ...]; see A010127.
For p = 47, we have [6; 1, 5, ...]; see A010137.
		

Links

• Piotr Miska, Maciej Ulas, On consecutive 1's in continued fractions expansions of square roots of prime numbers, arXiv:1904.03404 [math.NT], 2019. See Corollary 4.3. p. 13.
• Index entries for continued fractions for constants

Crossrefs

Cf. A040001, A010127, A010137, A307453.

Programs

PARI

isok(p) = my(cf = contfrac(sqrt(p))); (cf[2] == 1) && (cf[3] != 1);
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", ")));