A013647 -id:A013647 - 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

A099725 a(n) is the number of 1's in the period of the continued fraction of the square root of the n-th nonsquare integer.

Original entry on oeis.org

0, 1, 0, 0, 3, 1, 0, 0, 0, 4, 2, 1, 0, 0, 2, 0, 4, 2, 2, 1, 0, 0, 0, 2, 0, 4, 3, 2, 2, 1, 0, 0, 0, 0, 0, 0, 6, 6, 2, 6, 2, 1, 0, 0, 2, 2, 2, 0, 0, 4, 6, 2, 2, 4, 2, 1, 0, 0, 4, 0, 2, 2, 2, 0, 4, 4, 3, 6, 2, 2, 2, 1, 0, 0, 0, 2, 6, 0, 3, 0, 0, 5, 4, 6, 8, 2, 2, 8, 2, 1, 0, 0, 6, 0, 0, 4, 2, 4, 4, 0, 4, 4, 6, 2, 7

Offset: 1

Benoit Cloitre, Nov 07 2004

nonn

For sufficiently large period lengths, the fraction of 1's in the repeating part tends to log(4/3)/log(2) = 0.415... as from the Gauss-Kuzmin distribution, i.e., a(n) tends to 0.415...*A013943(n) for sufficiently large A013943(n). - A.H.M. Smeets, Jun 02 2018

The "n-th nonsquare integer" in the definition is A005117(n + 1). - Michael B. Porter, Jun 06 2018

A.H.M. Smeets, Table of n, a(n) for n = 1..10000

Cf. A005117, A013647, A013943.

from math import isqrt
from sympy.ntheory.continued_fraction import continued_fraction_periodic
def A099725(n): return (continued_fraction_periodic(0,1,n+(k:=isqrt(n))+int(n>=k*(k+1)+1))[-1]).count(1) # Chai Wah Wu, Jul 20 2024

cp's OEIS Frontend

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

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