A160520 - OEIS

%I #20 Jul 20 2024 15:33:53

%S 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,

%T 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,

%U 4,3,3,2,1,2,2

%N 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.

%e 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.

%t Length@Union@Last@ContinuedFraction@Sqrt@#&/@Select[Range@100,!IntegerQ@Sqrt@#&] (* _Giorgos Kalogeropoulos_, May 05 2022 *)

%o (Python)

%o import math

%o def findperiod( d ) :

%o     if len(d) == 0 :

%o         return 0

%o     for p in range( 1, len(d) - 1 ) :

%o         isPeriod = True

%o         for i in range(0, p) :

%o             for j in range(i + p, len(d), p ) :

%o                 if not d[i] == d[j] :

%o                     isPeriod = False

%o                     break

%o             if not isPeriod :

%o                 break

%o         if isPeriod :

%o             return len( set( d[:p] ) )

%o     return -1

%o def nextv( a, b, n, less = True ) :

%o     # print a, b, a[1]*a[1], n * a[0] * a[0]

%o     d = -1

%o     while (a[1]*a[1] < n * a[0] * a[0]) == less :

%o         d += 1

%o         a = ( a[0] + b[0], a[1] + b[1] )

%o     a = ( a[0] - b[0], a[1] - b[1] )

%o     return d, a, b

%o def generated( n ) :

%o     maxperiod = 100

%o     s = int( math.sqrt( n ) )

%o     if s * s == n :

%o         return []

%o     a = (1, 0)

%o     b = (0, 1)

%o     ds = []

%o     for i in range( maxperiod ):

%o         d, b, a = nextv( a, b, n )

%o         ds.append( d )

%o         d, b, a = nextv( a, b, n, less = False )

%o         ds.append( d )

%o     return ds[1:]

%o maxn = 1000

%o ns = [x for x in range( maxn ) if not int( math.sqrt( x ) ) ** 2 == x ]

%o v = list(map( findperiod, map( generated, ns ) ))

%o if v.count( -1 ) == 0 :

%o     print(v)

%o (Python)

%o from math import isqrt

%o from sympy.ntheory.continued_fraction import continued_fraction_periodic

%o 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

%Y Cf. A013943, A099725.

%K nonn

%O 1,2

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