A031779 -id:A031779 - cp's OEIS frontend

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

T. D. Noe, Feb 24 2012

nonn

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.

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

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).

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]

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

cp's OEIS Frontend

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

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