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

1,1

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).

Links

Table of n, a(n) for n=1..54.

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

Example

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.

Prog

(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, ", ")));

Crossrefs

Cf. A040001, A010127, A010137, A307453.

Sequence in context: A146142 A105854 A160022 * A187094 A060651 A146592

Adjacent sequences: A307527 A307528 A307529 * A307531 A307532 A307533

Keyword

nonn

Author

Michel Marcus, Apr 13 2019

Status

approved