A000924 -id:A000924 - cp's OEIS frontend

A035015 Period of continued fraction for square root of n-th squarefree integer.

1, 2, 1, 2, 4, 1, 2, 5, 4, 2, 1, 6, 6, 6, 4, 1, 5, 2, 8, 4, 4, 2, 1, 2, 2, 3, 2, 10, 12, 4, 2, 5, 4, 6, 7, 6, 11, 4, 1, 2, 10, 8, 6, 8, 7, 5, 6, 4, 4, 1, 2, 5, 10, 2, 5, 8, 10, 16, 4, 11, 1, 2, 12, 2, 9, 6, 15, 2, 6, 9, 6, 10, 10, 4, 1, 2, 12, 10, 3, 6, 16, 14, 9, 4, 18, 4, 4, 2, 1, 2, 9, 20, 10, 4

Offset: 2

David L. Treumann (alewifepurswest(AT)yahoo.com)

Friesen proved that each value appears infinitely often. - Michel Marcus, Apr 12 2019

			a(2)=1 because 2 is the 2nd smallest squarefree integer and sqrt 2 = [ 1,2,2,2,2,... ] thus has an eventual period of 1.

David W. Wilson, Table of n, a(n) for n = 2..10000
S. R. Finch, Class number theory [broken link]
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Christian Friesen, On continued fractions of given period, Proc. Amer. Math. Soc. 103 (1988), 9-14.
Ron Knott, An Introduction to Continued Fractions

Cf. A003285, A005117 (squarefree numbers), A013943.

sqf:= select(numtheory:-issqrfree,[$2..1000]):
map(n->nops(numtheory:-cfrac(sqrt(n),'periodic','quotients')[2]),sqf); # Robert Israel, Dec 21 2014

Length[ContinuedFraction[Sqrt[#]][[2]]]&/@Select[ Range[ 2,200], SquareFreeQ] (* Harvey P. Dale, Jul 17 2011 *)

a(n) = A003285(A005117(n)). - Michel Marcus, Dec 29 2014

Corrected and extended by James Sellers

A048941 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

Original entry on oeis.org

2, 4, 1, 10, 16, 2, 6, 20, 4, 3, 30, 8, 8, 34, 340, 4, 5, 394, 48, 10, 10, 52, 16, 5, 22, 3040, 6, 46, 70, 12, 12, 74, 50, 6, 64, 26, 6964, 20, 7, 48670, 96, 14, 14, 100, 36, 7, 970, 178, 30, 302, 198, 1060, 8, 39, 126, 16, 16, 130, 97684, 8, 25, 502, 6960, 34

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 A346419) give different results, with the first difference at n=14.
(End)
a(n) is the smallest positive integer x 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, A048942, 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((u2^2+D*v2^2)/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*u2+D*v1*v2)/q; \\ Jinyuan Wang, Sep 08 2021

Name edited by Michel Marcus, Jun 26 2020

Entry revised by Sean A. Irvine, Jul 13 2021

A053370 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of x for n == 1 mod 4.

Original entry on oeis.org

0, 1, 3, 2, 2, 19, 5, 27, 3, 131, 17, 7, 11, 943, 4, 4, 447, 13, 5035, 9, 37, 118, 703, 15371, 79, 1595, 87, 11, 28, 98, 10847, 6, 6, 57731, 604, 63, 1637147, 13, 478763, 20, 43331, 34, 3583111, 7, 7, 21639, 36, 66436843, 8011739, 872, 15, 5699, 77

Offset: 0

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039955.

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

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

Cf. A053370-A053375.

A053375 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 3 mod 4.

Original entry on oeis.org

1, 3, 3, 1, 39, 5, 273, 1, 4, 531, 7, 7, 12, 69, 5967, 413, 9, 9, 3, 165, 4, 22419, 93, 28, 105, 11, 11, 419775, 927, 6578829, 1, 140634693, 20, 105, 5019135, 13, 313191, 36, 123, 650783, 1, 1153080099, 4, 19162705353, 3, 5, 15, 15, 5, 3

Offset: 0

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039957.

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

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

Cf. A053370-A053375.

A039957 = Select[ Range[250], SquareFreeQ[#] && Mod[#, 4] == 3 & ]; A053375 = (NumberFieldFundamentalUnits[ Sqrt[#]][[1, 2, 2]] & ) /@ A039957 (* Jean-François Alcover, Jan 04 2012 *)

A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence).

Original entry on oeis.org

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 57, 73, 76

Offset: 1

N. J. A. Sloane

Euclidean fields that are not norm-Euclidean, such as Q(sqrt(14)) and Q(sqrt(69)), are not included. Actually, assuming GCH, a real quadratic field is Euclidean if and only if it is a PID (equivalently, if and only if it is a UFD). - Jianing Song, Jun 09 2022

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.

S. R. Finch, Class number theory [Cached copy, with permission of the author]
Erich Kaltofen and Heinrich Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Mathematics of Computation 53.188 (1989): 697-720. See page 698.
A. M. Odlyzko, Letters to N. J. A. Sloane Feb 1974
P. Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945-952.
Peter J. Weinberger, On Euclidean rings of algebraic integers, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 321-332.
Index entries for sequences related to quadratic fields

Cf. A003174, A003656.

A003174 = {2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}; Sort[ NumberFieldDiscriminant /@ Sqrt[A003174]] (* Jean-François Alcover, Jul 18 2012 *)

for(n=1,99,is_A003174(n) && print1(quaddisc(n)",")) \\ M. F. Hasler, Jan 26 2014

Equals A037449(A003174) as a set, not composition of functions (values are sorted by size; it turns out that a(n) is different from A037449(A003174(n)) for all n=1,...,16). - M. F. Hasler, Jan 26 2014

A006641 Class number of forms with discriminant -A003657(n), or equivalently class number of imaginary quadratic field with discriminant -A003657(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 2, 4, 2, 1, 5, 2, 2, 4, 4, 3, 1, 4, 7, 5, 3, 4, 6, 2, 2, 8, 5, 6, 3, 8, 2, 6, 10, 4, 2, 5, 5, 4, 4, 3, 10, 2, 7, 6, 4, 10, 1, 8, 11, 4, 5, 8, 4, 2, 13, 4, 9, 4, 3, 6, 14, 4, 7, 5, 4, 12, 2

Offset: 1

N. J. A. Sloane

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
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=1..3000
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Class Number

Cf. A003657.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric W. Weisstein *);
NumberFieldClassNumber@ Sqrt@ # & /@ Select[-Range@ 300, FundamentalDiscriminantQ]

for(n=1, 300, if(isfundamental(-n), print1(quadclassunit(-n).no, ", "))) \\ Andrew Howroyd, Jul 23 2018

[1] + [QuadraticField(-n, 'a').class_number() for n in (0..200) if is_fundamental_discriminant(-n) and not is_square(n)] # G. C. Greubel, Mar 01 2019

A014000 First coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 8, 2, 19, 5, 3, 27, 10, 3, 15, 131, 4, 17, 7, 11, 943, 170, 4, 4, 197, 447, 24, 13, 5035, 9, 5, 37, 118, 703, 11, 1520, 15371, 79, 35, 1595, 6, 87, 11, 28, 37, 25, 98, 10847, 6, 13, 3482, 6, 57731, 604, 24335, 63, 48, 1637147, 13, 478763

Offset: 2

Eric Rains (rains(AT)caltech.edu)

nonn

Taken from Cohen's table on pages 515-519. The table is indexed by the discriminant d = d(K) = A003658(n) of the real quadratic fields K. The fundamental unit is given as a pair of coordinates (a,b) = (A014000(n), A014046(n)) expressed in terms of the canonical integral basis (1,w) where w = (1+sqrt(d))/2 if d == 1 (mod 4), w = sqrt(d)/2 if d == 0 (mod 4).

The norm of this fundamental unit is A014077(n). The class number h(K) is A003652(n). - N. J. A. Sloane, Jun 14 2013

			Here is the start of Cohen's list of fundamental units: [0, 1], [1, 1], [2, 1], [1, 1], [3, 2], [2, 1], [5, 2], [8, 3], [2, 1], [19, 8], [5, 2], [3, 1], [27, 10], [10, 3], [3, 1], [15, 4], [131, 40],[4, 1], [17, 5], [7, 2], [11, 3], [943, 250], [170, 39], [4, 1], [4, 1], [197, 42], [447, 106], [24, 5], [13, 3], [5035, 1138], [9, 2], [5, 1], [37, 8], [118, 25], [703, 146], [11, 2], [1520, 273], [15371, 2968], [79, 15], [35, 6], [1595, 298], [6, 1], [87, 16], [11, 2], [28, 5], [37, 6], [25, 4], [98, 17], [10847, 1856], [6, 1], [13, 2], [3482, 531], [6, 1], [57731, 9384], [604, 97], [24335, 3588], [63, 10], [48, 7], [1637147, 253970], [13, 2], [478763, 72664], ... [_N. J. A. Sloane_, Jun 14 2013]

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.

S. R. Finch, Class number theory [Cached copy, with permission of the author]
Keith Matthews, Finding the fundamental unit of a real quadratic field
Index entries for sequences related to quadratic fields

Cf. A014046, A003658, A014077, A003652.

Edited by N. J. A. Sloane, Jun 14 2013

Offset corrected by Jianing Song, Mar 31 2019

A033197 Discriminants of quadratic number fields Q(sqrt -n) for n squarefree.

Original entry on oeis.org

-4, -8, -3, -20, -24, -7, -40, -11, -52, -56, -15, -68, -19, -84, -88, -23, -104, -116, -120, -31, -132, -136, -35, -148, -152, -39, -164, -168, -43, -184, -47, -51, -212, -55, -228, -232, -59, -244, -248, -260, -264, -67, -276, -280, -71, -292, -296, -308, -312, -79, -328, -83, -340, -344, -87, -356

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 103.

T. D. Noe, Table of n, a(n) for n = 1..10000

Values of n run through A005117. See A000924 for class numbers of these fields.

max = 56; j = 1; Do[ If[ SquareFreeQ[n], v[j] = n; j = j+1], {n, 1, 2*max}]; Do[ a[n] = -v[n]*If[Mod[v[n], 4] == 3, 1, 4], {n, 1, j-1}]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Oct 18 2011, after PARI *)

bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i:j=j+1)); M = vector(j-1); for (i=1,j-1,M[i]=if((L[i]%4==3),-L[i],-4*L[i])); M

For n squarefree and negative, a(n) = n if n == 1 (mod 4), otherwise a(n) = 4n.

A107997 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y odd.

Original entry on oeis.org

5, 13, 21, 29, 53, 61, 69, 77, 85, 93, 109, 133, 149, 157, 165, 173, 181, 205, 213, 221, 229, 237, 253, 277, 285, 293, 301, 309, 317, 341, 357, 365, 397, 413, 421, 429, 437, 445, 453, 461, 469, 493, 501, 509, 517, 533, 541, 565, 581, 589, 597, 613, 629, 645

Offset: 1

Steven Finch, Jun 13 2005

nonn

Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form (u + v*sqrt(m))/2, where u and v are both odd.

E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association Mathematical Tables, Vol. IV, London, 1934.
H. C. Williams, Eisenstein's problem and continued fractions, Utilitas Math. 37 (1990) 145-157.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 161 terms from Charles R Greathouse IV)
F. Arndt, Beiträge zur Theorie der quadratischen Formen, Archiv der Mathematik und Physik 15 (1850) 467-478.
A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental unit

Cf. A107998.

fQ[n_] := Block[{nffu = NumberFieldFundamentalUnits@ Sqrt@ n}, SquareFreeQ@ n && Denominator[ nffu[[1, 2, 2]]] > 1]; Select[ 8Range@ 81 - 3, fQ] (* Robert G. Wilson v, Dec 22 2014 *)

A003643 Number of genera of Q(sqrt(-n)), n squarefree.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 4, 2, 1, 2, 2, 4, 1, 4, 2, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 2, 2, 4, 2, 1, 2, 2, 4, 4, 1, 4, 4, 1, 2, 2, 4, 4, 1, 2, 1, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 1, 8, 2, 1, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 1, 4, 4, 1, 4, 2, 2, 4, 1, 4, 2, 2, 4, 2, 2, 1, 4, 2, 2, 2, 2, 4, 1, 8, 2, 1, 4, 2

Offset: 1

N. J. A. Sloane, Mira Bernstein

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for sequences related to quadratic fields

Cf. A000924, A001221 (omega), A005117, A033197, A003640.

Function[If[Mod[#, 4] == 1, 2^PrimeOmega[#], 2^(PrimeOmega[#] - 1)]] /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Sep 04 2019 *)

for(n=1, 200, if(issquarefree(n), print1(2^(omega(n*if((-n)%4>1, 4, 1)) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

a(n) = 2^(omega(A033197(n)) - 1). - Andrew Howroyd, Jul 24 2018

Let k = A005117(n) be the n-th squarefree number, then a(n) = 2^omega(k) if k == 1 (mod 4) and 2^(omega(k) - 1) otherwise. - Jianing Song, Jul 25 2018

cp's OEIS Frontend

A035015 Period of continued fraction for square root of n-th squarefree integer.

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A048941 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

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A053370 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of x for n == 1 mod 4.

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A053375 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 3 mod 4.

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Mathematica

A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence).

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Mathematica

PARI

Formula

A006641 Class number of forms with discriminant -A003657(n), or equivalently class number of imaginary quadratic field with discriminant -A003657(n).

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A014000 First coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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A033197 Discriminants of quadratic number fields Q(sqrt -n) for n squarefree.

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