A160520 a(n) is the number of distinct values in the periodic part of the continued fraction of the square root of n-th nonsquare positive integer.

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 4, 2, 3, 4, 3, 2, 1, 2, 3, 3, 2, 4, 2, 3, 3, 2, 1, 2, 2, 2, 2, 2, 4, 3, 3, 5, 3, 2, 1, 2, 4, 3, 4, 2, 2, 3, 2, 4, 3, 5, 3, 2, 1, 2, 5, 2, 4, 3, 4, 2, 3, 2, 2, 5, 4, 3, 3, 2, 1, 2, 2

Offset

1,2

Links

Table of n, a(n) for n=1..75.

Example

The 10th nonsquare positive integer is 13; sqrt(13) = cf[3;(1,1,1,1,6)], which has period 5, and 2 distinct values in its periodic part.

Mathematica

Length@Union@Last@ContinuedFraction@Sqrt@#&/@Select[Range@100, !IntegerQ@Sqrt@#&] (* Giorgos Kalogeropoulos, May 05 2022 *)

Prog

(Python)
import math
def findperiod( d ) :
    if len(d) == 0 :
        return 0
    for p in range( 1, len(d) - 1 ) :
        isPeriod = True
        for i in range(0, p) :
            for j in range(i + p, len(d), p ) :
                if not d[i] == d[j] :
                    isPeriod = False
                    break
            if not isPeriod :
                break
        if isPeriod :
            return len( set( d[:p] ) )
    return -1
def nextv( a, b, n, less = True ) :
    # print a, b, a[1]*a[1], n * a[0] * a[0]
    d = -1
    while (a[1]*a[1] < n * a[0] * a[0]) == less :
        d += 1
        a = ( a[0] + b[0], a[1] + b[1] )
    a = ( a[0] - b[0], a[1] - b[1] )
    return d, a, b
def generated( n ) :
    maxperiod = 100
    s = int( math.sqrt( n ) )
    if s * s == n :
        return []
    a = (1, 0)
    b = (0, 1)
    ds = []
    for i in range( maxperiod ):
        d, b, a = nextv( a, b, n )
        ds.append( d )
        d, b, a = nextv( a, b, n, less = False )
        ds.append( d )
    return ds[1:]
maxn = 1000
ns = [x for x in range( maxn ) if not int( math.sqrt( x ) ) ** 2 == x ]
v = list(map( findperiod, map( generated, ns ) ))
if v.count( -1 ) == 0 :
    print(v)
(Python)
from math import isqrt
from sympy.ntheory.continued_fraction import continued_fraction_periodic
def A160520(n): return len(set(continued_fraction_periodic(0, 1, n+(k:=isqrt(n))+int(n>=k*(k+1)+1))[-1])) # Chai Wah Wu, Jul 20 2024

Crossrefs

Cf. A013943, A099725.

Sequence in context: A092505 A066086 A323406 * A235708 A353929 A297770

Adjacent sequences: A160517 A160518 A160519 * A160521 A160522 A160523

Keyword

nonn

Author

Dremov Dmitry (dremovd(AT)gmail.com), May 16 2009

Status

approved