A039957 -id:A039957 - cp's OEIS frontend

A003641 Number of genera of imaginary quadratic field with discriminant -k, k = A039957(n).

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

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

In other words, this is the number of genera of those imaginary quadratic fields that have a discriminant which is odd and fundamental. The discriminant will be squarefree and of the form -4n+1. - Andrew Howroyd, Jul 25 2018

			a(4) = 2 because -15 = -A039957(4) and the number of genera of the quadratic field with discriminant -15 is 2. - _Andrew Howroyd_, Jul 25 2018

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).

Index entries for sequences related to quadratic fields

Cf. A001221 (omega), A003640, A003642, A039957.

2^(PrimeNu[Select[Range[1000], Mod[#, 4] == 3 && SquareFreeQ[#]&]] - 1) (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd *)

for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(2^(omega(n) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

a(n) = 2^(omega(A039957(n)) - 1). - Jianing Song, Jul 24 2018

Name clarified by Jianing Song, Jul 24 2018

A319660 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A039957(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0

Offset: 1

Jianing Song, Sep 25 2018

nonn

The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. In this case, G is the class group of Q(sqrt(-k)), and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(-k)) (cf. A003641).

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Cf. A003641, A039957, A319659, A319661.

for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(omega(n) - 1, ", ")))

a(n) = log_2(A003641(n)) = omega(A039957(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

A039956 Even squarefree numbers.

Original entry on oeis.org

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

A003657 Discriminants of imaginary quadratic fields, negated.

Original entry on oeis.org

3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 35, 39, 40, 43, 47, 51, 52, 55, 56, 59, 67, 68, 71, 79, 83, 84, 87, 88, 91, 95, 103, 104, 107, 111, 115, 116, 119, 120, 123, 127, 131, 132, 136, 139, 143, 148, 151, 152, 155, 159, 163, 164, 167, 168, 179, 183, 184, 187, 191

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Negative of 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. 223-234. See 4*A089269 = A191483 for even a(n) and A039957 for odd a(n). - Wolfdieter Lang, Nov 07 2003
All prime numbers in the set of the absolute values of negative fundamental discriminants are Gaussian primes (A002145). - Paul Muljadi, Mar 29 2008
Complement: 1, 2, 5, 6, 9, 10, 12, 13, 14, 16, 17, 18, 21, 22, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, ..., . - Robert G. Wilson v, Jun 04 2011
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, Feb 23 2021

Duncan A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989.
Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, p. 514.
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).

T. D. Noe, Table of n, a(n) for n=1..3000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See pp. 30, 53.
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, Dirichlet L-Series, Fundamental Discriminant.

Cf. A002145, A003658, A039957 (odd terms), A191483 (even terms), A104141.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *); -Select[-Range@ 194, FundamentalDiscriminantQ] (* Robert G. Wilson v, Jun 01 2011 *)

ok(n)={isfundamental(-n)} \\ Andrew Howroyd, Jul 20 2018

ok(n)={n<>1 && issquarefree(n/2^valuation(n,2)) && (n%4==3 || n%16==8 || n%16==4)} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..200) if is_fundamental_discriminant(-n)==1] # G. C. Greubel, Mar 01 2019

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

A039955 Squarefree numbers congruent to 1 (mod 4).

Original entry on oeis.org

1, 5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 129, 133, 137, 141, 145, 149, 157, 161, 165, 173, 177, 181, 185, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 249, 253, 257, 265, 269

Offset: 1

R. K. Guy

The subsequence of primes is A002144.
The subsequence of semiprimes (intersection with A001358) begins: 21, 33, 57, 65, 69, 77, 85, 93, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265.
The subsequence with more than two prime factors (intersection with A033942) begins: 105 = 3 * 5 * 7, 165 = 3 * 5 * 11, 273, 285, 345, 357, 385, 429, 465. - Jonathan Vos Post, Feb 19 2011
Except for a(1) = 1 these are the squarefree members of A079896 (i.e., squarefree determinants D of indefinite binary quadratic forms). - Wolfdieter Lang, Jun 01 2013
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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la formule t^2+a*u^2, a étant de la forme 4 n + 1, Essai sur la Théorie des Nombres An VI, Table IV. See first column. [_Paul Curtz_, Aug 14 2019]

Cf. A002144, A039956, A039957, A005117, A185197.

a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+1: n in [0..67] | IsSquarefree(4*n+1)];  // Bruno Berselli, Mar 03 2011

fQ[n_] := Max[Last /@ FactorInteger@ n] == 1 && Mod[n, 4] == 1; Select[ Range@ 272, fQ] (* Robert G. Wilson v *)
Select[Range[1,300,4],SquareFreeQ[#]&] (* Harvey P. Dale, Mar 27 2020 *)

list(lim)=my(v=List([1])); forfactored(n=5,lim\1, if(vecmax(n[2][,2])==1 && n[1]%4==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

is(n)=n%4==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

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 *)

A089269 Squarefree numbers congruent to 1 or 2 mod 4.

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 17, 21, 22, 26, 29, 30, 33, 34, 37, 38, 41, 42, 46, 53, 57, 58, 61, 62, 65, 66, 69, 70, 73, 74, 77, 78, 82, 85, 86, 89, 93, 94, 97, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 134, 137, 138, 141, 142, 145, 146, 149, 154, 157

Offset: 1

Wolfdieter Lang, Nov 07 2003

a(n) = one-fourth of the (negated) fundamental even discriminants D := b^2-4*a*c<0 of positive definite binary quadratic forms F=a*x^2+b*x*y+c*y^2. See A039957 for the odd numbers and A003657 for the combined even and odd numbers.

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

Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 231-234.
Arnold Scholz and Bruno Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Ch. 30.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000003, A005117, A039957, A003657, A185199.

[n: n  in [1..200] | IsSquarefree(n) and n mod 4 in [1,2]]; // Vincenzo Librandi, Oct 20 2017

Select[Range[200], MemberQ[{1, 2}, Mod[#, 4]]&& SquareFreeQ[#]&] (* Vincenzo Librandi, Oct 20 2017 *)

Entry revised by N. J. A. Sloane, May 28 2014

A191483 Even discriminants of imaginary quadratic fields, negated.

Original entry on oeis.org

4, 8, 20, 24, 40, 52, 56, 68, 84, 88, 104, 116, 120, 132, 136, 148, 152, 164, 168, 184, 212, 228, 232, 244, 248, 260, 264, 276, 280, 292, 296, 308, 312, 328, 340, 344, 356, 372, 376, 388, 404, 408, 420, 424, 436, 440, 452, 456, 472, 488, 516, 520, 532, 536, 548

Offset: 1

Robert G. Wilson v, Jun 04 2011

Terms of A003657 that are not in A039957.

The asymptotic density of this sequence is 1/Pi^2 (A092742). - Amiram Eldar, Feb 23 2021

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A003657, A039957, A089269, A092742.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *); -Select[-Range@550, FundamentalDiscriminantQ@# && EvenQ@# &]
(* Second program: *)
Select[Range[600], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&] (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd *)

ok(n)={isfundamental(-n) && n%2==0} \\ Andrew Howroyd, Jul 25 2018

ok(n)={n%4==0 && issquarefree(n/4) && n%16<>12} \\ Andrew Howroyd, Jul 25 2018

Complement(A003657, A039957).

a(n) = 4*A089269(n). - Andrew Howroyd, Jul 25 2018

A138610 Nonsquarefree numbers congruent to 3 mod 4.

Original entry on oeis.org

27, 63, 75, 99, 135, 147, 171, 175, 207, 243, 275, 279, 315, 343, 351, 363, 375, 387, 423, 459, 475, 495, 507, 531, 539, 567, 575, 603, 639, 675, 711, 735, 747, 775, 783, 819, 847, 855, 867, 875, 891, 927, 931, 963, 975, 999, 1035, 1071, 1075, 1083, 1107

Offset: 1

Zak Seidov, May 14 2008

nonn

Or, terms in A004767 but not in A039957.

The asymptotic density of this sequence is 1/4 - 2/Pi^2 = 0.047357... (A190357) - Amiram Eldar, Feb 10 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004767, A013929, A039957, A190357.

[4*n+3: n in [0..280] | not IsSquarefree(4*n+3)];  // Bruno Berselli, Mar 04 2011

remove(numtheory:-issqrfree, [seq(i,i=3..1200,4)]); # Robert Israel, Dec 18 2019

Mathematica
```
<Harvey P. Dale, Aug 27 2024 *)
```

A013929 INTERSECT A004767.

cp's OEIS Frontend

A003641 Number of genera of imaginary quadratic field with discriminant -k, k = A039957(n).

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A319660 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A039957(n).

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A039956 Even squarefree numbers.

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A003657 Discriminants of imaginary quadratic fields, negated.

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A230375 Squarefree numbers congruent to 2 or 3 mod 4.

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A039955 Squarefree numbers congruent to 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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