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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.

%I A160520 #20 Jul 20 2024 15:33:53
%S A160520 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 A160520 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 A160520 4,3,3,2,1,2,2
%N 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.
%e A160520 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 A160520 Length@Union@Last@ContinuedFraction@Sqrt@#&/@Select[Range@100,!IntegerQ@Sqrt@#&] (* _Giorgos Kalogeropoulos_, May 05 2022 *)
%o A160520 (Python)
%o A160520 import math
%o A160520 def findperiod( d ) :
%o A160520     if len(d) == 0 :
%o A160520         return 0
%o A160520     for p in range( 1, len(d) - 1 ) :
%o A160520         isPeriod = True
%o A160520         for i in range(0, p) :
%o A160520             for j in range(i + p, len(d), p ) :
%o A160520                 if not d[i] == d[j] :
%o A160520                     isPeriod = False
%o A160520                     break
%o A160520             if not isPeriod :
%o A160520                 break
%o A160520         if isPeriod :
%o A160520             return len( set( d[:p] ) )
%o A160520     return -1
%o A160520 def nextv( a, b, n, less = True ) :
%o A160520     # print a, b, a[1]*a[1], n * a[0] * a[0]
%o A160520     d = -1
%o A160520     while (a[1]*a[1] < n * a[0] * a[0]) == less :
%o A160520         d += 1
%o A160520         a = ( a[0] + b[0], a[1] + b[1] )
%o A160520     a = ( a[0] - b[0], a[1] - b[1] )
%o A160520     return d, a, b
%o A160520 def generated( n ) :
%o A160520     maxperiod = 100
%o A160520     s = int( math.sqrt( n ) )
%o A160520     if s * s == n :
%o A160520         return []
%o A160520     a = (1, 0)
%o A160520     b = (0, 1)
%o A160520     ds = []
%o A160520     for i in range( maxperiod ):
%o A160520         d, b, a = nextv( a, b, n )
%o A160520         ds.append( d )
%o A160520         d, b, a = nextv( a, b, n, less = False )
%o A160520         ds.append( d )
%o A160520     return ds[1:]
%o A160520 maxn = 1000
%o A160520 ns = [x for x in range( maxn ) if not int( math.sqrt( x ) ) ** 2 == x ]
%o A160520 v = list(map( findperiod, map( generated, ns ) ))
%o A160520 if v.count( -1 ) == 0 :
%o A160520     print(v)
%o A160520 (Python)
%o A160520 from math import isqrt
%o A160520 from sympy.ntheory.continued_fraction import continued_fraction_periodic
%o A160520 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 A160520 Cf. A013943, A099725.
%K A160520 nonn
%O A160520 1,2
%A A160520 Dremov Dmitry (dremovd(AT)gmail.com), May 16 2009