A003814 -id:A003814 - cp's OEIS frontend

A010333 Length of period of continued for sqrt(A003814(n)).

1, 1, 1, 5, 1, 1, 5, 1, 3, 1, 5, 7, 11, 1, 7, 5, 1, 5, 5, 11, 1, 9, 15, 9, 1, 5, 3, 9, 1, 9, 17, 1, 5, 21, 5, 13, 1, 7, 5, 1, 5, 11, 17, 7, 1, 9, 3, 7, 21, 13, 1, 5, 11, 17, 7, 11, 1, 19, 5, 5, 7, 15, 1, 5, 3, 5, 9, 21, 21, 1, 21, 37, 7, 21, 1, 5, 17, 25, 3, 15, 13, 1, 9, 19, 19, 1, 5, 7, 39

Offset: 1

N. J. A. Sloane, Walter Gilbert

nonn

If the period length of the continued fraction of a quadratic surd sqrt(D) is odd, the negative Pell equation x^2 - D y^2 = -1 has (infinite) solutions.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A003814.

Select[Map[Length@ Last@ ContinuedFraction@ Sqrt@ # &, Range[550]], OddQ] (* Michael De Vlieger, Dec 06 2017, after T. D. Noe at A003814 *)

A031396 Numbers k such that Pell equation x^2 - k*y^2 = -1 is soluble.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 290, 293, 298

Offset: 1

David W. Wilson

nonn

Terms are divisible neither by 4 nor by a prime of the form 4*k + 3 (although these conditions are not sufficient - compare A031398). - Lekraj Beedassy, Aug 17 2005

This is the set of integer solutions of all quadratic forms m^2*x^2 -/+ b*x + c with discriminant b^2 - 4*m^2*c = -4 where m is any term of A004613. - Klaus Purath, Jun 18 2025

Harvey Cohn, "Advanced Number Theory".

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Dmitry Berdinsky and Prohrak Kruengthomya, Nonstandard Cayley automatic representations of fundamental groups, arXiv:2001.04743 [math.GR], 2020.
Dmitry Berdinsky and Prohrak Kruengthomya, Nonstandard Cayley Automatic Representations for Fundamental Groups of Torus Bundles over the Circle, International Conference on Language and Automata Theory and Applications (LATA 2020): Language and Automata Theory and Applications, Lecture Notes in Computer Science, Vol 12038. Springer, Cham, 115-127.
Hsin-Te Chiang, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu, and Chiun-Chang Lee, On negative Pell equations: Solvability and unsolvability in integers, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 10-26.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
D. Khurana and T. Y. Lam, Invertible commutators in matrix rings, J. Algebra and Applications, 10 (211), 51-71.
K. Lakshmi and R. Someshwari, On The Negative Pell Equation y^2 = 72x^2 - 23, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).
Morris Newman, A note on an equation related to the Pell equation, The American Mathematical Monthly 84.5 (1977): 365-366.
R. Suganya and D. Maheswari, On the Negative Pellian Equation y^2 = 110 * x^2 - 29, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.
A. Vijayasankar, M. A. Gopalan, and V. Krithika, On The Negative Pell Equation y^2 = 112 * x^2 - 7, International Journal of Engineering and Applied Sciences (IJEAS 2017), Vol. 4, Issue 7, 11-14.

Equals {1} U A003814.
Cf. A031398, A002313, A133204, A130226 (values of x).
See also A322781, A323271, A323272.

fQ[n_] := Solve[x^2 + 1 == n*y^2, {x, y}, Integers] != {}; Select[ Range@ 300, fQ] (* Robert G. Wilson v, Dec 19 2013 *)

def is_A031396(k):
    if k==1: return True
    if Integer(k).is_square(): return False
    K. = QuadraticField(k)
    return continued_fraction(a).period_length()%2
print([k for k in range(1, 1000) if is_A031396(k)])  # Robin Visser, Nov 02 2024

A031423 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 10.

Original entry on oeis.org

701, 1418, 1493, 2197, 2290, 3257, 4793, 6154, 6466, 8389, 8753, 9577, 9965, 10765, 11257, 11677, 12541, 14218, 14929, 15413, 15658, 16001, 16501, 17009, 17786, 18049, 18314, 18581, 19121, 21577, 22157, 22745, 24557, 24677, 25805, 26561, 27530, 28517

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe and Georg Fischer)

Cf. A031404-A031422.

Subsequence of A003814.

n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 10 && c[[2, (len + 1)/2 - 1]] == 10, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 04 2014; corrected by Georg Fischer, Jun 23 2019 *)

from sympy.ntheory.continued_fraction import continued_fraction_periodic
A031423_list = []
for n in range(1,10**4):
    cf = continued_fraction_periodic(0,1,n)
    if len(cf) > 1 and len(cf[1]) > 1 and len(cf[1]) % 2 and cf[1][len(cf[1])//2] == 10:
        A031423_list.append(n) # Chai Wah Wu, Sep 16 2021

a(1) corrected by T. D. Noe, Apr 04 2014

a(1) = 26 removed by Georg Fischer, Jun 23 2019

A172000 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm -1.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 137, 145, 148, 149, 153, 157, 160, 162, 164, 170, 173, 180, 181, 185, 193, 197, 200

Offset: 1

Max Alekseyev, Jan 21 2010

nonn

Complement of A087643 in the nonsquare integers A000037.
Subsequence of A000415, their set difference form A172001.
Contains A003814 as a subsequence, their squarefree terms coincide and form A003654.
It seems that this sequence also gives the values of n such that the equation x^2 - n*y^2 = n has integer solutions. - Colin Barker, Aug 20 2013

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == -1, AppendTo[cr, n]]], {n, 2, 1000}]; cr (* Artur Jasinski, Oct 10 2011 *)

{ for(n=1,1000, if(issquare(n),next); if( norm(bnfinit(x^2-n).fu[1])==-1, print1(n,", ")) ) }

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A003654.

Edited by Max Alekseyev, Mar 09 2010

A003654 Squarefree integers m such that the fundamental unit of Q(sqrt(m)) has norm -1. Also, squarefree integers m such that the Pell equation x^2 - m*y^2 = -1 is soluble.

Original entry on oeis.org

2, 5, 10, 13, 17, 26, 29, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 313, 314, 317, 337, 346, 349, 353, 362

Offset: 1

N. J. A. Sloane, Mira Bernstein. Entry revised by N. J. A. Sloane, Jun 11 2012

nonn

The squarefree elements of A003814 and A172000. - Max Alekseyev, Jun 01 2009
Together with {1} and A031398 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - Max Alekseyev, Mar 09 2010
Squarefree integers m such that Q(sqrt(m)) contains the infinite continued fraction [k, k, k, k, k, ...] for some positive integer k. For example, Q(sqrt(5)) contains [1, 1, 1, 1, 1, ...] which equals (1 + sqrt(5))/2. - Greg Dresden, Jul 23 2010

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 46.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 56.
W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 1..9446
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
P. Seeling, Ueber die Aufloesung der Gleichung x^2-Ay^2=+-1 in ganzen Zahlen, wo A positiv und kein vollstaendiges Quadrat sein muss, Archiv der Mathematik und Physik, Vol. 52 (1871), p. 40-49.

Cf. A031396, A031397, A003814, A249021.

isA003654 := proc(n)
    local cf,p ;
    if not numtheory[issqrfree](n) then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        if modp(p,4) = 3 then
            return false;
        end if;
    end do:
    cf := numtheory[cfrac](sqrt(n),'periodic','quotients') ;
    type( nops(op(2,cf)),'odd') ;
end proc:
A003654 := proc(n)
    option remember;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA003654(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003654(n),n=1..40) ; # R. J. Mathar, Oct 19 2014

Reap[For[n = 2, n < 1000, n++, If[SquareFreeQ[n], sol = Solve[x^2 - n y^2 == -1, {x, y}, Integers]; If[sol != {}, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Mar 24 2020 *)

Edited by Max Alekseyev, Mar 17 2010

A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 233, 241, 257, 265, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 365, 373, 389, 397, 401, 409, 421, 425, 433, 445, 449, 457, 461, 481, 485

Offset: 1

Wolfdieter Lang, Nov 29 2002

Nonsquare integers n for which Pell equation x^2 - n*y^2 = -4 has infinitely many integer solutions. The smallest solutions are given in A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, table p. 108).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Pell's equation

A subsequence of A077425.
Odd elements of A003814.
Cf. A077427, A172000.

isOddPrim := proc(n::integer)
    local cf;
    cf := numtheory[cfrac]((sqrt(n)+1)/2,'periodic','quotients') ;
    if nops(op(2,cf)) mod 2 =1 then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
notA077426 := proc(n::integer)
    if issqr(n) then
        RETURN(true) ;
    elif not isOddPrim(n) then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
A077426 := proc(n::integer)
    local resul,i ;
    resul := 5 ;
    i := 1 ;
    while i < n do
        resul := resul+4 ;
        while notA077426(resul) do
            resul := resul+4 ;
        od ;
        i:= i+1 ;
    od ;
    RETURN(resul) ;
end:
for n from 1 to 61 do print(A077426(n)) ; od : # R. J. Mathar, Apr 25 2006

fQ[n_] := !IntegerQ@ Sqrt@ n && OddQ@ Length@ ContinuedFraction[(Sqrt@ n + 1)/2][[2]]; Select[Range@ 500, fQ] (* Robert G. Wilson v, Nov 17 2012 *)

Edited and extended by Max Alekseyev, Mar 03 2010

Edited by Max Alekseyev, Mar 05 2010

A077232 a(n) is smallest natural number satisfying Pell equation a^2 - d(n)*b^2= +1 or = -1, with d(n)=A000037(n) (a nonsquare). Corresponding smallest b(n)=A077233(n).

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 3, 10, 7, 18, 15, 4, 4, 17, 170, 9, 55, 197, 24, 5, 5, 26, 127, 70, 11, 1520, 17, 23, 35, 6, 6, 37, 25, 19, 32, 13, 3482, 199, 161, 24335, 48, 7, 7, 50, 649, 182, 485, 89, 15, 151, 99, 530, 31, 29718, 63, 8, 8, 65, 48842, 33, 7775, 251, 3480, 17, 1068, 43, 26, 57799, 351, 53, 80, 9, 9, 82, 55, 378, 10405, 28, 197, 500, 19, 1574, 1151, 12151, 2143295, 39, 49, 5604, 99, 10, 10, 101, 227528

Offset: 1

Wolfdieter Lang, Nov 08 2002

If d(n)=A000037(n) is from A003654 (that is if the regular continued fraction for sqrt(d(n)) has odd (primitive) period length) then the -1 option applies. For such d(n) the minimal a(n) and b(n) numbers for the +1 option are 2*a(n)^2+1 and 2*a(n)*b(n), respectively (see Perron I, pp. 94,95).
If d(n)=A000037(n)= k^2+1, k=1,2,.., then the a^2 - d(n)*b^2 = -1 Pell equation has the minimal solution a(n)=k and b(n)=1. If d(n)=A000037(n)= k^2-1, k=2,3,..., then the a^2 - d(n)*b^2 = +1 Pell equation has the minimal solution a=k and b=1.
The general integer solutions (up to signs) of Pell equation a^2 - d(n)*b^2 = +1 with d(n)=A000037(n), but not from A003654, are a(n,p)= T(p+1,a(n)) and b(n,p)= b(n)*S(p,2*a(n)), p=0,1,... If d(n)=A000037(n) is also from A003654 then these solutions are a(n,p)= T(p+1,2*a(n)^2+1) and b(n,p)= 2*a(n)*b(n)*S(p,2*(2*a(n)^2+1)), p=0,1,... Here T(n,x), resp. S(n,x) := U(n,x/2), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
The general integer solutions (up to signs) of the Pell equation a^2 - d(n)*b^2 = -1 with d(n)=A000037(n)= A003654(k), for some k>=1, are a(n,p) = a(n)*(S(n,2*(2*a(n)^2)+1) + S(n-1,2*(2*a(n)^2)+1)) and b(n,p) = b(n)*(S(n,2*(2*a(n)^2)+1) - S(n-1,2*(2*a(n)^2)+1)) with the S(n,x) := U(n,x/2) Chebyshev polynomials. S(-1,x) := 0.
If the trivial solution x=1, y=0 is included, the sequence becomes A006702. - T. D. Noe, May 17 2007

			d=10=A000037(7)=A003654(3), therefore a(7)^2=10*b(7)^2 -1, i.e. 3^2=10*1^2 -1 and 2*a(7)^2+1=19 and 2*a(7)*b(7)=2*3*1=6 satisfy 19^2 - 10*6^2 = +1.
d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) solution of the a^2 - d*b^2 = -1 Pell equation and a(8)=10 and b(8)=A077233(8)=3 satisfy 10^2 - 11*3^2 = +1.
10=d(7)=A000037(7)=A003654(3)=3^2+1 hence a(7)=3 and b(7)=1 are the smallest numbers satisfying a^2-10*b^2=-1.
8=d(6)=A000037(6)=3^2-1 (not in A003654) hence a(6)=3 and b(6)=1 are the smallest numbers satisfying a^2-8*b^2=+1.

T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, table p. 301.
O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 26, p. 91 with explanation on pp. 94,95).

Ray Chandler, Table of n, a(n) for n = 1..10000
A. M. Legendre, Fractions les plus simples m/n qui satisfont à l'équation m^2 - an^2 =+-1 pour tout nombre non quarré a depuis 2 jusqu'à 1003, Essai sur la Théorie des Nombres An VI, Table XII. [_Paul Curtz_, Apr 10 2019]
Index entries for sequences related to Chebyshev polynomials.

Cf. A000037, A003654, A003814, A006702, A033313, A077233.

nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
nonSquare[n_] := n + Round[Sqrt[n]];
a[n_] := a[n] = Module[{lr}, lr = FindLinearRecurrence[ Numerator[ Convergents[ Sqrt[nonSquare[n]], nconv]]]; (1/2) SelectFirst[lr, #>1&]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 10 2021 *)

a(n)=sqrt(A000037(n)*A077233(n)^2 + (-1)^(c(n))) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.

A077233 a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 1, 3, 2, 5, 4, 1, 1, 4, 39, 2, 12, 42, 5, 1, 1, 5, 24, 13, 2, 273, 3, 4, 6, 1, 1, 6, 4, 3, 5, 2, 531, 30, 24, 3588, 7, 1, 1, 7, 90, 25, 66, 12, 2, 20, 13, 69, 4, 3805, 8, 1, 1, 8, 5967, 4, 936, 30, 413, 2, 125, 5, 3, 6630, 40, 6, 9, 1, 1, 9, 6, 41, 1122, 3, 21, 53, 2, 165, 120, 1260, 221064, 4, 5, 569, 10, 1, 1, 10, 22419

Offset: 1

Wolfdieter Lang, Nov 08 2002

If d(n)=A000037(n) is from A003654 (that is if the regular continued fraction for sqrt(d(n)) has odd (primitive) period length) then the -1 option applies. For such d(n) the minimal b(n) and a(n) numbers for the +1 option are 2*b(n)^2 + 1 and 2*b(n)*a(n), respectively (see Perron I, pp. 94,p5).
For general integer solutions see A077232 comments.
If the trivial solution x=1, y=0 is included, the sequence becomes A006703. - T. D. Noe, May 17 2007

			d=10=A000037(7)=A003654(3), therefore a(7)=1 and b(7)=A077232(7)=3 give 3^2=10*1^2 -1 and 2*b(7)^2+1=19 and 2*b(7)*a(7)=2*3*1=6 satisfy 19^2 - 10*6^2 = +1.
d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) solution of the b^2 - d*a^2 = -1 Pell equation and a(8)=3 and b(8)=A077232(8)=10 satisfy 10^2 - 11*3^2 = +1. See A077232 for further examples.

T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, table p. 301.
O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 26, p. 91 with explanation on pp. 94,95).

Ray Chandler, Table of n, a(n) for n = 1..10000
A. M. Legendre, Fractions les plus simples m/n qui satisfont à l'équation m^2 - an^2 =+-1 pour tout nombre non quarré a depuis 2 jusqu'à 1003, Essai sur la Théorie des Nombres An VI, Table XII. [_Paul Curtz_, Apr 10 2019]

Cf. A000037, A003654, A003814, A033317, A077232.

nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
nonSquare[n_] := n + Round[Sqrt[n]];
b[n_] := b[n] = Module[{lr}, lr = FindLinearRecurrence[ Numerator[ Convergents[ Sqrt[nonSquare[n]], nconv]]]; (1/2) SelectFirst[lr, #>1&]];
a[n_] := If[n == 1, 1, SelectFirst[{Sqrt[(b[n]^2 - 1)/nonSquare[n]], Sqrt[(b[n]^2 + 1)/nonSquare[n]]}, IntegerQ]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 10 2021 *)

a(n)=sqrt((A077232(n)^2 - (-1)^(c(n)))/A000037(n)) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.

A206586 Numbers k such that the periodic part of the continued fraction of sqrt(k) has positive even length.

Original entry on oeis.org

3, 6, 7, 8, 11, 12, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 34, 35, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

T. D. Noe, Mar 19 2012

nonn

By making the length positive, we exclude squares. See A206587 for this sequence and the squares. All primes of the form 4m + 3 are here.

T. D. Noe, Table of n, a(n) for n = 1..1000
P. J. Rippon and H. Taylor, Even and odd periods in continued fractions of square roots, Fibonacci Quarterly 42, May 2004, pp. 170-180.

Cf. A003814 (period is odd), A206587.

Select[Range[100], ! IntegerQ[Sqrt[#]] && EvenQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &]

cyc(cf) = {
  if(#cf==1, return(0)); \\ There is no cycle
  my(s=[]);
  for(k=2, #cf,
    s=concat(s, cf[k]);
    if(cf[k]==2*cf[1], return(s)) \\ Cycle found
  );
  0 \\ Cycle not found
}
select(n->(t=#cyc(contfrac(sqrt(n))))>0 && t%2==0, vector(100, n, n)) \\ Colin Barker, Oct 19 2014

A031397 Nonsquarefree n such that Pell equation x^2 - n y^2 = -1 is soluble.

Original entry on oeis.org

50, 125, 250, 325, 338, 425, 845, 925, 1025, 1250, 1325, 1445, 1450, 1525, 1625, 1682, 1825, 1850, 2050, 2125, 2197, 2425, 2725, 2738, 2825, 2873, 2890, 3050, 3125, 3250, 3425, 3625, 3725, 3925, 4250, 4325, 4394, 4625, 4825, 4901, 4913

Offset: 1

David W. Wilson

nonn

Harvey Cohn, Advanced Number Theory, Dover Publications, New York, N.Y. (1980).
S Vidhyalakshmi, V Krithika, K Agalya, On The Negative Pell Equation, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 2, February (2016) www.ijeter.everscience.org,

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 500 terms from Robert Israel)

Equals {A003814} minus {A003654}, cf. A031396.

filter:= t -> not numtheory:-issqrfree(t) and [isolve(x^2 - t*y^2 = -1)]<>[]:
select(filter, [$1..10000]); # Robert Israel, Jul 10 2018

r[n_] := Reduce[x>0 && y>0 && x^2 - n y^2 == -1, {x, y}, Integers];
Reap[For[n = 1, n <= 5000, n++, If[!SquareFreeQ[n], If[r[n] =!= False, Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Mar 05 2019 *)

Offset changed by Robert Israel, Jul 10 2018

cp's OEIS Frontend

A010333 Length of period of continued for sqrt(A003814(n)).

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A031396 Numbers k such that Pell equation x^2 - k*y^2 = -1 is soluble.

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A031423 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 10.

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A172000 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm -1.

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A003654 Squarefree integers m such that the fundamental unit of Q(sqrt(m)) has norm -1. Also, squarefree integers m such that the Pell equation x^2 - m*y^2 = -1 is soluble.

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A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

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A077232 a(n) is smallest natural number satisfying Pell equation a^2 - d(n)*b^2= +1 or = -1, with d(n)=A000037(n) (a nonsquare). Corresponding smallest b(n)=A077233(n).

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A077233 a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).

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