A000924 -id:A000924 - cp's OEIS frontend

A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, 42, 5, 1, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3, 6630, 40, 6, 9

Offset: 1

Eric W. Weisstein

nonn

D = D(n) = A000037(n). - Wolfdieter Lang, Oct 04 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Laurent Beeckmans, Squares Expressible as Sum of Consecutive Squares, Am. Math. Monthly, Volume 101, Number 5, page 442, May 1994.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column B page 19.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Pell Equation

Cf. A000037, A033313 (for the x's), A077232, A077233.

F:= proc(d) local r,Q; uses numtheory;
  Q:= cfrac(sqrt(d),'periodic','quotients'):
  r:= nops(Q[2]);
  if r::odd then
    denom(cfrac([op(Q[1]),op(Q[2]),op(Q[2][1..-2])]))
  else
    denom(cfrac([op(Q[1]),op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr,[$1..100])); # Robert Israel, May 17 2015

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033317 = DeleteCases[PellSolve /@ Range[100], {}][[All, 2]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002349 *)

a(n) = sqrt((A033313(n)^2 - 1)/A000037(n)). - Jinyuan Wang, Jul 09 2020

A078355 Minimal (positive) solution a(n) of Pell equation b(n)^2 - D(n)*a(n)^2 = +4 with D(n)= A077425(n). The companion sequence is a(n)=A077428(n).

Original entry on oeis.org

1, 3, 16, 1, 5, 8, 24, 640, 1, 7, 40, 195, 32, 3, 534000, 1, 9, 106000, 3, 12754704, 40, 8, 6525, 226592, 1, 11, 2968, 15, 1039424, 16, 48, 305, 352, 3621, 1856, 1, 13, 9384, 126585, 1360, 8, 896073208080, 56, 72664, 3, 6440, 5, 521904, 1, 15, 140510608, 5

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

For the conversion of the (x,y) values of Perron's table to the (b(n),a(n)) values see a A077428 comment.

For the general solution of Pell b^2 - D(n)*a^2 = +4 see a comment in A077428 (with a and b interchanged).

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

d = Select[Range[5, 300, 4], !IntegerQ[Sqrt[#]]&]; a[n_] := Module[{a, b, r}, b /. {r = Reduce[a > 0 && b > 0 && a^2 - d[[n]]*b^2 == 4, {a, b}, Integers]; (r /. C[1] -> 0) || (r /. C[1] -> 1) // ToRules} // Select[#, IntegerQ, 1] &] // First; Table[a[n], {n, 1, 52}] (* Jean-François Alcover, Jul 30 2013 *)

More terms from Max Alekseyev, Mar 03 2010

A087048 Class numbers of indefinite quadratic forms over the integers in two variables with discriminant D = D(n) = A079896(n), n>=1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Aug 07 2003

nonn

An indefinite quadratic form over the integers in two variables F(x,y) := a*x^2 + b*x*y + c*y^2 has discriminant D := b^2 - 4*a*c >0 not a square (a and c non-vanishing); that is D=D(n)= A079896(n) = [5,8,12,13,17,20,21,...], n>=1.
For a given discriminant D from A079896(n) a reduced form [a,b,c] is defined by b>0 and f(D)-min(|2*a|,|2*c|) <= b < f(D), with f(D) := ceiling(sqrt(D)).
For a given discriminant D from A079896(n) every primitive reduced form [a,b,c] defines a periodic chain of such forms by applying repeatedly the transformation R(t)*[a,b,c]=[a'(t),b'(t),c'(t)]=[c,-b+2*c*t,F(-1,t)] with uniquely defined t= ceiling((f(D)+b)/(2*c))-1 if c>0 and t=-(ceiling((f(D)+b)/(2*|c|)-1)) if c<0. The number of such (different) periodic chains of primitive reduced forms is called the class number for this (indefinite) discriminant D from A079896(n). - Wolfdieter Lang, Jun 07 2013
A primitive form [a,b,c] has gcd(a,b,c)=1.
See the Appendix 2 of the Buell reference. pp. 235-243, for the class numbers, called H(D), for the fundamental discriminants 0 < D < 10000. Table 2A gives the class numbers for squarefree D == 1 (mod 4) and Table 2B the ones for D == 0 (mod 4), with  D/4 squarefree and not congruent to 1 modulo 4 (compare Buell, p. 69, 1. and 2.). - Wolfdieter Lang, May 29 2013
For an online program for D < 10^6 see the Keith Matthews link. - Wolfdieter Lang, Jul 24 2019

			n=3, D(3) = A079896(3) = 12, a(3) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (both with period length 2): [[-2, 2, 1], [1, 2, -2]] and [[-1, 2, 2], [2, 2, -1]].
n=14, D(14) = A079896(14) = 40, a(14) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (with period length 6 resp. 2): [[-3, 2, 3], [3, 4, -2], [-2, 4, 3], [3, 2, -3], [-3, 4, 2], [2, 4, -3]] and  [[-1, 6, 1], [1, 6, -1]].
n=36, D(36) = A079896(36) = 89, a(36) = 1 because there is only one periodic chain of primitive reduced forms [a,b,c] (with period length 14): [[ -5, 3, 4], [4, 5, -4], [-4, 3, 5], [5, 7, -2], [-2, 9, 1], [1, 9, -2], [-2, 7, 5], [5, 3, -4], [-4, 5, 4], [4, 3, -5], [-5, 7, 2], [2, 9, -1], [-1, 9, 2], [2, 7, -5]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the form [1, 9, -2].
n=62, D(62) = A079896(62) = 148, a(62) = 3 because there are three periodic chains of primitive reduced forms [a,b,c] (with period length 6 and 6 and 2, resp.): [[-7, 6, 4], [4, 10, -3], [-3, 8, 7], [7, 6, -4], [-4, 10, 3], [3, 8, -7]] and [[-4, 6, 7], [7, 8, -3], [-3, 10, 4], [4, 6, -7], [-7, 8, 3], [3, 10, -4]] and [[-1, 12, 1], [1, 12, -1]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the forms [4, 10, -3] and [3, 10, -4] and [1, 12, -1], resp.

D. A. Buell, Binary Quadratic Forms, Springer, 1989.
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch. 31, pp. 112 ff.

Robin Visser, Table of n, a(n) for n = 1..10000
S. R. Finch, Class number theory [broken link]
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Wolfdieter Lang, Table of n-1, D(n), a(n) for n=1, ..., 136
Keith Matthews, Finding the class number h(d) of primitive binary quadratic forms of positive discriminant d

See A006374 for another version. Cf. A079896.

def a(n):
    i, D, S = 1, Integer(5), []
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            if gcd([a, b, (D-b^2)/(4*a)]) > 1: continue
            Q = BinaryQF(a, b, -(D-b^2)/(4*a))
            if all([(not Q.is_equivalent(t)) for t in S]): S.append(Q)
    return len(S)  # Robin Visser, May 31 2025

Offset corrected by Robin Visser, May 31 2025

A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).

Original entry on oeis.org

5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 245, 249, 253, 257

Offset: 1

Wolfdieter Lang, Nov 29 2002

The Pell equation x^2 - a(n)*y^2 = +4 has infinitely many (integer) solutions (see A077428 and A078355).
These are the odd numbers in A079896. The even ones are 4*A000037. - Wolfdieter Lang, Sep 15 2015
First differences: 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 8, ... , only 4's and 8's?. - Paul Curtz, Apr 11 2019
Yes. There are only 4's and 8's. Proof: Only multiples of 4 may appear. The 4's correspond to successive composite in A016813, whereas an 8 corresponds to a square. A greater multiple of 4 would imply to have at least 2 consecutive squares in A016813, which is not possible since 2 consecutive squares cannot have a difference of 4. That sequence of 4's and 8's can be obtained with A010052 (without the 1st term) where the 0's are replaced with 4's and 1's replaced with 8's. - Michel Marcus, Apr 16 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. R. Finch, Class number theory [Cached copy, with permission of the author]
A. M. Legendre, Expression les plus simples des formules Ly^2+Myz+Nz^2 où M est impair pour toutes les valeurs de B = M^2-4LN depuis B=5 jusqu'à B=305, Essai sur la Théorie des Nombres An VI, Table II. [_Paul Curtz_, Apr 11 2019]

Intersection of A016813 and A000037.

Cf. A077426, A079896.

A077425 := proc(n::integer) local resul,i ; resul := 5 ; i := 1 ; while i < n do resul := resul+4 ; while issqr(resul) do resul := resul+4 ; od ; i:= i+1 ; od ; RETURN(resul) ; end proc:
seq(A077425(n),n=1..31) ; # R. J. Mathar, Apr 25 2006

Select[Range[5,300,4],!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Dec 05 2012 *)

[n | n <- vector(100,n,4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014

list(lim)=my(v=List()); for(s=2,sqrtint((lim\=1)+1), forstep(n=s^2 + if(s%2,4,1), min((s+1)^2-1,lim), 4, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Nov 04 2021

from operator import sub
from sympy import integer_nthroot
def A077425(n): return n+sub(*integer_nthroot(n,2))<<2|1 # Chai Wah Wu, Oct 01 2024

More terms from Max Alekseyev, Mar 03 2010

A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73

Offset: 1

N. J. A. Sloane

These integers yield norm-Euclidean real quadratic fields. There are other positive integers, e.g., D=14 or D=69, for which Q[sqrt(D)] is Euclidean, but for a Euclidean function different from the field norm.

For further references see sequence A048981 which also lists negative D corresponding to (complex) norm-Euclidean fields. - M. F. Hasler, Jan 26 2014

H. Cohn, A Second Course in Number Theory, Wiley, NY, 1962, p. 109.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 213.
K. Inkeri, Über den Euklidischen Algorithmus in quadratischen Zahlkörpern. Ann. Acad. Sci. Fennicae Ser. A. 1. Math.-Phys., No. 41, 1-35, 1947. [Incorrectly gives 97 as a member of this sequence.]
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 2, p. 57.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.

H. Chatland and H. Davenport, Euclid's algorithm in real quadratic fields, Canadian J. Math. 2, (1950), 289-296.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Pierre Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945-952.
Index entries for sequences related to quadratic fields

Cf. A003173, A003246, A048981, A187776, A263465.

is_A003174(n) = bittest(9444877083272958060780,n) \\ M. F. Hasler, Jan 26 2014

a(n) = A048981(n+5). - M. F. Hasler, Jan 26 2014

Definition corrected and comment rephrased by M. F. Hasler, Jan 26 2014

Definition corrected by Jonathan Sondow, Oct 19 2015

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

A037449 Discriminant of quadratic field Q(sqrt(n)).

Original entry on oeis.org

1, 8, 12, 1, 5, 24, 28, 8, 1, 40, 44, 12, 13, 56, 60, 1, 17, 8, 76, 5, 21, 88, 92, 24, 1, 104, 12, 28, 29, 120, 124, 8, 33, 136, 140, 1, 37, 152, 156, 40, 41, 168, 172, 44, 5, 184, 188, 12, 1, 8, 204, 13, 53, 24, 220, 56, 57, 232, 236, 60, 61, 248, 28, 1, 65, 264, 268, 17, 69

Offset: 1

Jason Earls, Jun 30 2001

For the discriminant of the quadratic field Q(sqrt(-n)), see A204993.

a(n) is the smallest positive N such that (n/k) = (n/(k mod N)) for every odd k that is coprime to n, where (n/k) is the Jacobi symbol. As we have Dirichlet's theorem on arithmetic progressions, a(n) is also the smallest positive N such that (n/p) = (n/(p mod N)) for every odd prime p that is not a factor of n. - Jianing Song, May 16 2024

T. D. Noe, Table of n, a(n) for n = 1..1000
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A204993, A007913.

Table[NumberFieldDiscriminant[Sqrt[n]], {n, 100}] (* Artur Jasinski, Jan 27 2012 *)

PARI
```
vector(150,n,quaddisc(n))
```

[fundamental_discriminant(n) for n in (1..69)] # Peter Luschny, Oct 15 2018

Let b(n) = A007913(n), then a(n) = b(n) if b(n) == 1 (mod 4) and 4*b(n) otherwise. - Jianing Song, May 16 2024

A003649 Class number of real quadratic field Q(sqrt f), where f is the n-th squarefree number A005117(n).

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane and Mira Bernstein

nonn

Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 322-326, Theorem 12.6.1, Example 12.6.7, Table 8.
D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, p. 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..6083 (squarefree numbers < 10000)
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Index entries for sequences related to quadratic fields

Cf. A000924.

DeleteCases[Table[Boole[FreeQ[FactorInteger[n], {, k /; k > 2}]] * NumberFieldClassNumber[Sqrt[n]], {n, 100}], 0] (* Alonso del Arte, Aug 26 2014 *)

for(n=2,1e3,if(issquarefree(n),print1(qfbclassno(n*if(n%4>1, 4, 1))", "))) \\ Charles R Greathouse IV, Feb 19 2013

A048942 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

Original entry on oeis.org

2, 2, 1, 4, 6, 1, 2, 6, 1, 1, 8, 2, 2, 8, 78, 1, 1, 84, 10, 2, 2, 10, 3, 1, 4, 546, 1, 8, 12, 2, 2, 12, 8, 1, 10, 4, 1062, 3, 1, 7176, 14, 2, 2, 14, 5, 1, 132, 24, 4, 40, 26, 138, 1, 5, 16, 2, 2, 16, 11934, 1, 3, 60, 826, 4, 250, 10, 6, 39, 1, 12, 18, 2, 2, 18

Offset: 1

Eric W. Weisstein

nonn

From Sean A. Irvine, Jul 16 2021: (Start)
These values are computed by Algorithm 5.7.2 in Cohen.
Other methods of computation (see A346420) give different results, with the first difference at n=14.
(End)
a(n) is the smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = +-4, where D = A000037(n). - Jinyuan Wang, Sep 08 2021

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993.

Jinyuan Wang, Table of n, a(n) for n = 1..1000
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Fundamental Unit.

Cf. A000037, A007913, A048941, A346419, A346420.

a(n) = my(A, D=n+(1+sqrtint(4*n))\2, d=sqrtint(D), p, q, t, u1, u2, v1, v2); if(d%2==D%2, p=d, p=d-1); u1=-p; u2=2; v1=1; v2=0; q=2; while(v2==0 || q!=t, A=(p+d)\q; t=p; p=A*q-p; if(t==p && v2!=0, return(2*u2*v2/q), t=A*u2+u1; u1=u2; u2=t; t=A*v2+v1; v1=v2; v2=t; t=q; q=(D-p^2)/q)); (u1*v2+u2*v1)/q; \\ Jinyuan Wang, Sep 08 2021

Name edited by Michel Marcus, Jun 26 2020

Entry revised by Sean A. Irvine, Jul 16 2021

A003652 Class number of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Mira Bernstein

nonn

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..3001
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Class Number

NumberFieldClassNumber[Sqrt[#]] &/@ Select[Range[500], FundamentalDiscriminantQ] (* G. C. Greubel, Mar 01 2019 *)

for(n=1, 500, if(isfundamental(n) && !issquare(n), print1(quadclassunit(n).no, ", "))) \\ G. C. Greubel, Mar 01 2019

[QuadraticField(n, 'a').class_number() for n in (1..500) if is_fundamental_discriminant(n) and not is_square(n)] # G. C. Greubel, Mar 01 2019

Offset corrected by Jianing Song, Mar 31 2019

cp's OEIS Frontend

A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

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A078355 Minimal (positive) solution a(n) of Pell equation b(n)^2 - D(n)*a(n)^2 = +4 with D(n)= A077425(n). The companion sequence is a(n)=A077428(n).

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A087048 Class numbers of indefinite quadratic forms over the integers in two variables with discriminant D = D(n) = A079896(n), n>=1.

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A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).

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A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.

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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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A037449 Discriminant of quadratic field Q(sqrt(n)).

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