A206578 The least number with exactly n ones in the continued fraction of its square root.

2, 3, 14, 7, 13, 91, 43, 115, 94, 819, 133, 1075, 211, 1219, 309, 871, 421, 1147, 244, 3427, 478, 2575, 991, 8791, 604, 3799, 886, 5539, 1381, 8851, 1279, 7303, 1561, 19519, 1759, 10339, 1831, 12871, 2038, 13771, 1999, 8611, 1516, 15871, 2731, 20875, 1726

Offset

0,1

Comments

It appears that only the odd-numbered terms 3 and 7 are prime; all other primes occur at even-numbered terms 0, 4, 6, 12, 16, 22, 28, 30, 34, ... In terms 0 to 1000, there are 268 primes and 632 semiprimes.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Mathematica

nn = 50; zeros = nn; t = Table[0, {nn}]; k = 2; While[zeros > 0, If[! IntegerQ[Sqrt[k]], cnt = Count[ContinuedFraction[Sqrt[k]][[2]], 1]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = k; zeros--]]; k++]; Join[{2}, t]

Prog

(Python)
from sympy import continued_fraction_periodic
def A206578(n):
    m = 1
    while True:
        s = continued_fraction_periodic(0, 1, m)[-1]
        if isinstance(s, list) and s.count(1) == n:
            return m
        m += 1 # Chai Wah Wu, Jun 12 2017

Crossrefs

Cf. A013647-A013650 (0-3), A020440-A020446 (4-10), A031779-A031868 (11-100).

Cf. A206582 (n twos), A206583 (n threes), A206584 (n fours), A206585 (n fives).

Sequence in context: A287858 A288648 A287912 * A056435 A244295 A032806

Adjacent sequences: A206575 A206576 A206577 * A206579 A206580 A206581

Keyword

nonn

Author

T. D. Noe, Feb 24 2012

Status

approved