A039955 -id:A039955 - cp's OEIS frontend

A039956 Even squarefree numbers.

2, 6, 10, 14, 22, 26, 30, 34, 38, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 202, 206, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262

Offset: 1

R. K. Guy

Sum of even divisors = 2 * the sum of odd divisors. - Amarnath Murthy, Sep 07 2002
From Daniel Forgues, May 27 2009: (Start)
a(n) = n * (3/1) * zeta(2) + O(n^(1/2)) = n * (3/1) * (Pi^2 / 6) + O(n^(1/2)).
For any prime p_i, the n-th squarefree number even to p_i (divisible by p_i) is:
n * ((p_i + 1)/1) * zeta(2) + O(n^(1/2)) = n * ((p_i + 1)/1) * (Pi^2 / 6) + O(n^(1/2)).
For any prime p_i, there are as many squarefree numbers having p_i as a factor as squarefree numbers not having p_i as a factor amongst all the squarefree numbers (one-to-one correspondence, both cardinality aleph_0).
E.g., there are as many even squarefree numbers as there are odd squarefree numbers.
For any prime p_i, the density of squarefree numbers having p_i as a factor is 1/p_i of the density of squarefree numbers not having p_i as a factor.
E.g., the density of even squarefree numbers is 1/p_i = 1/2 of the density of odd squarefree numbers (which means that 1/(p_i + 1) = 1/3 of the squarefree numbers are even and p_i/(p_i + 1) = 2/3 are odd) and as a consequence the n-th even squarefree number is very nearly p_i = 2 times the n-th odd squarefree number (which means that the n-th even squarefree number is very nearly (p_i + 1) = 3 times the n-th squarefree number while the n-th odd squarefree number is very nearly (p_i + 1)/ p_i = 3/2 the n-th squarefree number).
(End)
Apart from first term, these are the tau2-atoms as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019

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

T. D. Noe, Table of n, a(n) for n = 1..10000
D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
G. J. O. Jameson, Even and odd square-free numbers, Math. Gazette 94 (2010), 123-127; Author's copy.
James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.

Subsequence of A005117.

Cf. A002808, A056911, A039955, A039957, A056911, A092673, A264387.

a039956 n = a039956_list !! (n-1)
a039956_list = filter even a005117_list  -- Reinhard Zumkeller, Aug 15 2011

[n: n in [2..262 by 2] | IsSquarefree(n)];  // Bruno Berselli, Mar 03 2011

select(numtheory:-issqrfree,[seq(i,i=2..1000,4)]); # Robert Israel, Dec 23 2015

Select[Range[2,270,2],SquareFreeQ] (* Harvey P. Dale, Jul 23 2011 *)

is(n)=n%4==2 && issquarefree(n) \\ Charles R Greathouse IV, Sep 13 2013

Numbers k such that A092673(k) = +- 2. - Jon Perry, Mar 02 2004
Sum_{n>=1} 1/a(n)^s = zeta(s)/((1+2^s)*zeta(2*s)). - Enrique Pérez Herrero, Sep 15 2012 [corrected by Amiram Eldar, Sep 26 2023]
a(n) = 2*A056911(n). - Robert Israel, Dec 23 2015
a(n) = 2*(1+2*A264387(n)),  n >= 1. - Wolfdieter Lang, Dec 24 2015

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

Original entry on oeis.org

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197

Offset: 1

N. J. A. Sloane, Mira Bernstein, Eric W. Weisstein

All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008

Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - Charles R Greathouse IV, Jan 21 2022

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3001 from T. D. Noe)
Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.
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, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Fundamental Discriminant.
Eric Weisstein's World of Mathematics, Class Number.

Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.

Union of A039955 and 4*A230375.

fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* Jean-François Alcover, Nov 02 2011, after Eric W. Weisstein *)
Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &]  (* Alonso del Arte, Apr 02 2014, based on Arkadiusz Wesolowski's program for A094612 *)
max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* Alonso del Arte, Apr 02 2014 *)

v=[]; for(n=1,500,if(isfundamental(n),v=concat(v,n))); v

list(lim)=my(v=List()); forsquarefree(n=1,lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v,n[1]))); Set(v) \\ Charles R Greathouse IV, Jan 21 2022

def is_fundamental(d):
    r = d % 4
    if r > 1 : return False
    if r == 1: return (d != 1) and is_squarefree(d)
    q = d // 4
    return is_squarefree(q) and (q % 4 > 1)
[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018

Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).

a(n) ~ (Pi^2/3)*n. There are (3/Pi^2)*x + O(sqrt(x)) terms up to x. - Charles R Greathouse IV, Jan 21 2022

More terms from Eric W. Weisstein and Jason Earls, Jun 19 2001

A039957 Squarefree numbers congruent to 3 mod 4.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 31, 35, 39, 43, 47, 51, 55, 59, 67, 71, 79, 83, 87, 91, 95, 103, 107, 111, 115, 119, 123, 127, 131, 139, 143, 151, 155, 159, 163, 167, 179, 183, 187, 191, 195, 199, 203, 211, 215, 219, 223, 227, 231, 235, 239, 247, 251, 255

Offset: 1

R. K. Guy

Negatives of odd fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 224-230. See 4*A089269 = A191483 for the even case and A003657 for combined even and odd numbers. - Wolfdieter Lang, Nov 07 2003

The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la forme t^2+au^2 a étant un nombre de la forme 4n-1, Essai sur la Théorie des Nombres An VI, Table V.

Cf. A039955, A039956, A185197, A191483.

a039957 n = a039957_list !! (n-1)
a039957_list = filter ((== 3) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+3: n in [0..63] | IsSquarefree(4*n+3)];  // Bruno Berselli, Mar 04 2011

fQ[n_] := SquareFreeQ[n] && Mod[n, 4] == 3; Select[ Range@ 258, fQ] (* Robert G. Wilson v, Mar 02 2011 *)
Select[Range[3,300,4],SquareFreeQ] (* Harvey P. Dale, Mar 08 2015 *)

is(n)=n%4==3 && issquarefree(n) \\ Charles R Greathouse IV, Feb 07 2017

Offset corrected

A230375 Squarefree numbers congruent to 2 or 3 mod 4.

Original entry on oeis.org

2, 3, 6, 7, 10, 11, 14, 15, 19, 22, 23, 26, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 51, 55, 58, 59, 62, 66, 67, 70, 71, 74, 78, 79, 82, 83, 86, 87, 91, 94, 95, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122, 123, 127, 130, 131, 134, 138, 139, 142, 143

Offset: 1

Colin Barker, Oct 17 2013

nonn

Also numbers such that the discriminant of the quadratic field Q(sqrt(n)) equals 4n. - Michel Marcus, Nov 26 2013

The asymptotic density of this sequence is 4/Pi^2 (A185199). - Amiram Eldar, Feb 23 2021

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Quadratic field.

Cf. A039955, A039957, A185199.

Select[Range[200],SquareFreeQ[#]&&MemberQ[{2,3},Mod[#,4]]&] (* Harvey P. Dale, Feb 04 2015 *)

s=[]; for(n=1, 200, if(issquarefree(n) && n%4!=1, s=concat(s, n))); s

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.

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Original entry on oeis.org

1, 1, 2, 1, 1, 8, 2, 10, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 2968, 15, 298, 16, 2, 5, 17, 1856, 1, 1, 9384, 97, 10, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 1084152, 117, 2, 746, 10, 88, 157, 126890, 1, 1, 1311, 56, 287

Offset: 1

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039955.

Subsequence of A077058 excluding terms for which A077425(n) is not squarefree. - Max Alekseyev, Dec 12 2012

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

Emmanuel Vantieghem, 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. A053370-A053375.

2*NumberFieldFundamentalUnits[ Sqrt[#] ][[1, 2, 2]] & /@ Select[ Range[5, 309, 4], SquareFreeQ ]  (* Jean-François Alcover, Jul 09 2013 *)

forstep(n=5,1000,4, if(!issquarefree(n),next); print1( 2*polcoeff(lift(bnfinit(x^2-n).fu[1]),1), ", " )) /* Max Alekseyev */

A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

Original entry on oeis.org

17, 33, 37, 41, 57, 65, 73, 89, 97, 101, 105, 113, 129, 137, 141, 145, 161, 177, 185, 193, 197, 201, 209, 217, 233, 241, 249, 257, 265, 269, 273, 281, 305, 313, 321, 329, 337, 345, 349, 353, 373, 377, 381, 385, 389, 393, 401, 409, 417, 433, 449, 457, 465, 473, 481, 485, 489, 497, 505, 521, 537, 545, 553, 557, 561, 569, 573

Offset: 1

Emmanuel Vantieghem, Mar 07 2017

nonn

Squarefree integers m congruent to 1 modulo 4 such that the minimal solution of the Pell equation x^2 - d*y^2 = +-4 has both x and y even.
The sequence contains the squarefree numbers congruent to 5 modulo 8 that are not in A107997.
This sequence union A107997 = A039955.
This sequence contains all numbers of the form 4*k^2+1 (k > 1) that are squarefree.

			33 is in the sequence since the fundamental unit of the field Q(sqrt(33)) is 23+4*sqrt(33).
53 is not in the sequence since the fundamental unit of the field Q(sqrt(53)) is 3+omega, where omega = (1+sqrt(53))/2.

Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

Keith Matthews, Finding the fundamental unit of a real quadratic field

Cf. A107997, A039955, A107998, A197170.

cp's OEIS Frontend

A039956 Even squarefree numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

Extensions

A039957 Squarefree numbers congruent to 3 mod 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Extensions

A230375 Squarefree numbers congruent to 2 or 3 mod 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

Views

Author

Keywords

Comments

Examples