A035015 -id:A035015 - cp's OEIS frontend

A003285 Period of continued fraction for square root of n (or 0 if n is a square).

0, 1, 2, 0, 1, 2, 4, 2, 0, 1, 2, 2, 5, 4, 2, 0, 1, 2, 6, 2, 6, 6, 4, 2, 0, 1, 2, 4, 5, 2, 8, 4, 4, 4, 2, 0, 1, 2, 2, 2, 3, 2, 10, 8, 6, 12, 4, 2, 0, 1, 2, 6, 5, 6, 4, 2, 6, 7, 6, 4, 11, 4, 2, 0, 1, 2, 10, 2, 8, 6, 8, 2, 7, 5, 4, 12, 6, 4, 4, 2, 0, 1, 2, 2, 5, 10, 2, 6, 5, 2, 8, 8, 10, 16, 4, 4, 11, 4, 2, 0, 1, 2, 12

Offset: 1

N. J. A. Sloane

Any string of five consecutive terms m^2 - 2 through m^2 + 2 for m > 2 in the sequence has the corresponding periods 4,2,0,1,2. - Lekraj Beedassy, Jul 17 2001
For m > 1, a(m^2+m) = 2 and the continued fraction is m, 2, 2*m, 2, 2*m, 2, 2*m, ... - Arran Fernandez, Aug 14 2011
Apparently the generating function of the sequence for the denominators of continued fraction convergents to sqrt(n) is always rational and of form p(x)/[1 - C*x^m + (-1)^m * x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+m) - (-1)^m * b(n), where a(n) is equal to m for each nonsquare n, or 0. See A006702 for the conjecture regarding C. The same conjectures apply to the sequences of the numerators of continued fraction convergents to sqrt(n). - Ralf Stephan, Dec 12 2013
If a(n)=1, n is of form k^2+1 (A002522 except the initial term 1). See A013642 for a(n)=2, A013643 for a(n)=3, A013644 for a(n)=4, A010337 for a(n)=5, A020347 for a(n)=6, A010338 for a(n)=7, A020348 for a(n)=8, A010339 for a(n)=9, and furthermore A020349-A020439. - Ralf Stephan, Dec 12 2013
From William Krier, Dec 12 2024: (Start)
a(m^2-4) = 4 for even m>=6 since sqrt(m^2-4) = [m-1; 1, (m-4)/2, 1, 2*(m-1)].
a(m^2-4) = 6 for odd m>=5 since sqrt(m^2-4) = [m-1; 1, (m-3)/2, 2, (m-3)/2, 1, 2*(m-1)].
a(m^2+4) = 2 for even m>=2 since sqrt(m^2+4) = [m; m/2, 2*m].
a(m^2+4) = 5 for odd m>=3 since sqrt(m^2+4) = [m; (m-1)/2, 1, 1, (m-1)/2, 2*m]. (End)

A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 197.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)
Marius Beceanu, Period of the Continued Fraction of sqrt(n), (Feb 05 2003)
Leon Bernstein, Fundamental units and cycles in the period of real quadratic number fields, I. Pacific J. Math 63 (1976): 37-61.
Ron Knott, All square-root continued fractions eventually repeat
R. Luczak, Patterns in the period lengths of simple periodic continued fractional representations of square roots of integers near perfect squares, QED: Chicago's Youth Math Research Symposium (April 2013).
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Justin T. Miller, Families of Continued Fractions, translated (2000) from a document of Nikos Drakos, Computer Based Learning Unit, University of Leeds.
C. D. Patterson and H. C. Williams, Some Periodic Continued Fractions with Long Periods, Mathematics of Computation, Vol. 44 (1985), No. 170, pp. 523-532.
A. M. Rockett and P. Szuesz, On the lengths of the periods of the continued fractions of square-roots of integers, Forum Mathematicum, 2 (1990), 119-123.
R. G. Stanton, C. Sudler, Jr. and H. C. Williams, An Upper Bound for the Period of the Simple Continued Fraction for Sqrt(D), Pacific Journal of Math., Vol. 67 (1976), No. 2, pp. 525-536.
Hanna Uscka-Wehlou, Continued Fractions and Digital Lines with Irrational Slopes, in Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Volume 4992/2008, Springer-Verlag.
A. J. van der Poorten, Fractional Parts of the Period of the Continued Fraction Expansion of Quadratic Integers [Refined and revised text of a talk given at the 2nd Conference of the Canadian Number Theory Association, Vancouver, 1989]
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
H. C. Williams, A Numerical Investigation Into the Length of the Period of the Continued Fraction Expansion of Sqrt(D), Mathematics of Computation, Vol. 36 (1981), No. 154, pp. 593-601.

Cf. A013943, A035015, A054269, A061490, A065938, A067280, A097853.

f:= n ->  if issqr(n) then 0
   else nops(numtheory:-cfrac(sqrt(n),'periodic','quotients')[2]) fi:
map(f, [$1..100]); # Robert Israel, Sep 02 2015

a[n_] := ContinuedFraction[Sqrt[n]] // If[Length[ # ] == 1, 0, Length[Last[ # ]]]&
pcf[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],0,Length[ContinuedFraction[s][[2]]]]]; Array[pcf,110] (* Harvey P. Dale, Jul 15 2017 *)

a(n)=if(issquare(n),return(0));my(s=sqrt(n),x=s,f=floor(s),P=[0],Q=[1],k);while(1,k=#P;P=concat(P,f*Q[k]-P[k]);Q=concat(Q,(n-P[k+1]^2)/Q[k]);k++;for(i=1,k-1,if(P[i]==P[k]&&Q[i]==Q[k],return(k-i)));x=(P[k]+s)/Q[k];f=floor(x)) \\ Charles R Greathouse IV, Jul 31 2011

isok(n, p) = {localprec(p); my(cf = contfrac(sqrt(n))); setsearch(Set(cf), 2*cf[1]);}
a(n) = {if (issquare(n), 0, my(p=100); while (! isok(n, p), p+=100); localprec(p); my(cf = contfrac(sqrt(n))); for (k=2, #cf, if (cf[k] == 2*cf[1], return (k-1))););} \\ Michel Marcus, Jul 07 2021

from sympy.ntheory.continued_fraction import continued_fraction_periodic
def a(n):
    cfp = continued_fraction_periodic(0, 1, d=n)
    return 0 if len(cfp) == 1 else len(cfp[1])
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Aug 22 2021

A013943 Period of continued fraction for sqrt(m), m = n-th nonsquare.

Original entry on oeis.org

1, 2, 1, 2, 4, 2, 1, 2, 2, 5, 4, 2, 1, 2, 6, 2, 6, 6, 4, 2, 1, 2, 4, 5, 2, 8, 4, 4, 4, 2, 1, 2, 2, 2, 3, 2, 10, 8, 6, 12, 4, 2, 1, 2, 6, 5, 6, 4, 2, 6, 7, 6, 4, 11, 4, 2, 1, 2, 10, 2, 8, 6, 8, 2, 7, 5, 4, 12, 6, 4, 4, 2, 1, 2, 2, 5, 10, 2, 6, 5, 2, 8, 8, 10, 16, 4, 4, 11, 4, 2, 1, 2, 12, 2, 2, 9, 6, 8, 15, 2, 6, 6

Offset: 1

Clark Kimberling

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Kival Ngaokrajang, Illustration of initial terms, periodic are colored.
Eric Weisstein's World of Mathematics, Continued Fraction.

Cf. A003285, A035015.

nonSquares = Select[Range[120], !IntegerQ[Sqrt[#]]&]; a[n_] := Length[ Last[ ContinuedFraction[ Sqrt[ nonSquares[[n]] ]]]]; Table[a[n], {n, 1, Length[nonSquares]}] (* Jean-François Alcover, May 27 2013 *)

from math import isqrt
from sympy.ntheory.continued_fraction import continued_fraction_periodic
def A013943(n): return len(continued_fraction_periodic(0,1,n+(k:=isqrt(n))+int(n>=k*(k+1)+1))[-1]) # Chai Wah Wu, Jul 20 2024

A107356 Period of continued fraction for (1 + square root of n-th squarefree integer)/2.

Original entry on oeis.org

2, 2, 1, 4, 4, 2, 2, 1, 4, 2, 3, 6, 2, 6, 4, 2, 1, 2, 8, 4, 4, 2, 3, 6, 6, 5, 4, 10, 8, 4, 2, 1, 4, 6, 6, 6, 3, 4, 3, 6, 10, 4, 6, 8, 9, 6, 2, 4, 4, 2, 2, 1, 6, 2, 7, 8, 2, 12, 4, 9, 3, 6, 12, 6, 18, 6, 7, 4, 6, 7, 6, 6, 14, 4, 2, 2, 12, 10, 6, 6, 4, 10, 7, 4, 18, 4, 4, 2, 3, 6, 5, 20, 14, 8, 5, 12, 6, 10

Offset: 1

Steven Finch, May 24 2005

nonn

			a(7) = 2 because 11 is the 7th smallest squarefree integer and (1 + sqrt 11)/2 = [2,6,3,6,3,6,3,... ] thus has an eventual period of 2. We omit 1 from the list of squarefree integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, An Introduction to Continued Fractions
K. S. Williams and N. Buck, Comparison of the lengths of the continued fractions of sqrt(D) and (1/2)*(1+sqrt(D)), Proc. Amer. Math. Soc. 120 (1994) 995-1002.

Cf. A005117, A035015, A146326.

(* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) s = Drop[ Select[ Range[162], SquareFreeQ[ # ] &], 1]; Length[ ContinuedFraction[ # ][[2]]] & /@ ((1 + Sqrt[s])/2) (* Robert G. Wilson v, May 27 2005 *)
Length[ContinuedFraction[(Sqrt[#]+1)/2][[2]]]&/@Select[Range[ 2,200], SquareFreeQ] (* Harvey P. Dale, Aug 16 2021 *)

a(n) = A146326(A005117(n+1)). - R. J. Mathar, Sep 24 2009

More terms from Robert G. Wilson v, May 27 2005

A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

Original entry on oeis.org

2, 1, 1, 2, 2, 6, 3, 3, 2, 1, 5, 7, 1, 6, 2, 10, 5, 2, 8, 4, 2, 1, 7, 6, 4, 11, 2, 13, 8, 2, 7, 7, 4, 7, 20, 9, 11, 2, 9, 8, 19, 2, 6, 6, 21, 20, 1, 2, 2, 18, 9, 9, 16, 3, 21, 12, 3, 12, 2, 27, 11, 10, 18, 3, 34, 13, 17, 2, 8, 23, 12, 5, 18, 2, 22, 11, 24, 15, 26, 15, 6, 22, 27, 2, 31, 4, 2

Offset: 2

Robin Visser, Aug 21 2025

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.

			For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) is of the form u or u*(2 + sqrt(2)), for some totally positive unit u. Thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) is a totally positive unit, so a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) is a totally positive unit, so a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) is of the form u or u*(3 + sqrt(6)), for some totally positive unit u. Thus a(5) = 2.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387207.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0,2*s+1,2) for t in range(cf[i+1])]
    for i in range(len(ind)):
        for j in range(i):
            if ((ind[i]/ind[j]) in OK) and (OK(ind[i]/ind[j]).is_unit()): break
        else: ans += 1
    return ans

A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

Original entry on oeis.org

2, 1, 1, 3, 2, 10, 5, 3, 2, 1, 4, 9, 1, 11, 2, 26, 7, 6, 10, 4, 2, 1, 9, 19, 13, 10, 7, 21, 9, 2, 25, 13, 9, 7, 58, 29, 15, 2, 16, 33, 33, 3, 14, 10, 18, 74, 1, 3, 2, 82, 41, 21, 43, 13, 22, 30, 7, 18, 5, 24, 25, 51, 34, 4, 106, 53, 27, 11, 37, 28, 57, 9, 59, 2, 122, 61, 42, 16, 130, 65, 11

Offset: 2

Robin Visser, Aug 21 2025

			For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) has norm either 1 or 2, thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) has norm 1, thus a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) has norm 1, thus a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) has norm either 1 or 3, thus a(5) = 3.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387203.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0, 2*s+1, 2) for t in range(cf[i+1])]
    return max([c.norm() for c in ind])

a(n) <= A005117(n) for all n >= 2 [Dress-Scharlau].

cp's OEIS Frontend

A003285 Period of continued fraction for square root of n (or 0 if n is a square).

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A013943 Period of continued fraction for sqrt(m), m = n-th nonsquare.

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A107356 Period of continued fraction for (1 + square root of n-th squarefree integer)/2.

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A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

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A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

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