A133675 -id:A133675 - cp's OEIS frontend

A033195 Erroneous version of A133675.

3, 4, 7, 8, 11, 16, 19, 27, 28, 43, 67, 163

Offset: 1

dead

Included in accordance with OEIS policy of listing published but incorrect sequences, with pointers to the correct versions.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 260.

Cf. A133675.

A032354 j-invariants for orders of class number 1.

Original entry on oeis.org

0, 1728, -3375, 8000, -32768, 54000, 287496, -884736, -12288000, 16581375, -884736000, -147197952000, -262537412640768000

Offset: 0

N. J. A. Sloane

This sequence may also be written as [0, 12^3, -15^3, 20^3, -32^3, 2*30^3, 66^3, -96^3, -3*160^3, 255^3, -960^3, -5280^3, -640320^3]. Compare A267195. - N. J. A. Sloane, Jan 27 2016

H. Cohn, Introduction to the Construction of Class Fields, Cambridge; p. 183.
D. A. Cox, Primes of the form x^2+ny^2, Wiley, p. 261.

See A267195 for (essentially) the cube roots of these numbers.

See A133675 (times -1) for the corresponding discriminants.

A133288 Negative discriminants with form class group of exponent 2 (negated).

Original entry on oeis.org

15, 20, 24, 32, 35, 36, 40, 48, 51, 52, 60, 64, 72, 75, 84, 88, 91, 96, 99, 100, 112, 115, 120, 123, 132, 147, 148, 160, 168, 180, 187, 192, 195, 228, 232, 235, 240, 267, 280, 288, 312, 315, 340, 352, 372, 403, 408, 420, 427, 435, 448, 480, 483, 520, 532, 555, 595, 627, 660, 672, 708, 715, 760

Offset: 1

David Brink, Dec 30 2007

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989.
D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Andrew Howroyd, Table of n, a(n) for n = 1..88

Cf. A003171 is the disjoint union of A133675 and this sequence.

ok(n)={(-n)%4<2 && quadclassunit(-n).no > 1 && !#select(k->k<>2, quadclassunit(-n).cyc)} \\ Andrew Howroyd, Jul 20 2018

A322710 Negative discriminants with form class number 2 (negated).

Original entry on oeis.org

15, 20, 24, 32, 35, 36, 40, 48, 51, 52, 60, 64, 72, 75, 88, 91, 99, 100, 112, 115, 123, 147, 148, 187, 232, 235, 267, 403, 427

Offset: 1

Jianing Song, Dec 24 2018

This is the full sequence.

The j-invariants for these discriminants are quadratic integers. See the links below for a full list.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013
Jianing Song, j-invariants for the discriminants with form class number 2 and their Hilbert class polynomials
Eric Weisstein's World of Mathematics, Class Number

Cf. A014603, A305474.

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), this sequence (isomorphic to C_2), A328825 (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

for(n=1, 500, if((-n)%4<=1&&quadclassunit(-n)[1]==2, print1(n, ", ")))

A328825 Negative discriminants with form class group isomorphic to C_3 (negated).

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 92, 107, 108, 124, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

Jianing Song, Dec 05 2019

Also negative discriminants with form class number 3.
Conjecture: this sequence is finite and this is the full list.
The fundamental terms are listed in A006203, and that is a full sequence.
From Jianing Song, May 17 2021: (Start)
Equivalently, negative discriminants of orders whose class group is isomorphic to C_3 (negated).
The known even terms are all congruent to 12 modulo 16. Among the known even terms, k/4 is either here or in A133675. What's the reason for that?
Among the known terms, k is in A023679 if and only if k is in this sequence and k/4 is not. Is there a connection between these two sequences? (End)

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

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), this sequence (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

Cf. A006203, A023679.

isA328825(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[3] \\ Corrected by Jianing Song, May 17 2021

A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 3, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 4, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 5, 4, 0, 0, 2, 2, 0, 0, 3, 4, 0

Offset: 1

T. D. Noe, May 18 2005, Apr 30 2008

nonn

This sequence is closely related to the class number function, h(-n), which is given for fundamental discriminants in A006641. For a fundamental discriminant d, we have h(-d) < 2a(d). It appears that a(n) < Sqrt(n) for all n. For k>1, the primes p for which a(p)=k coincide with the numbers n such that the class number h(-n) is 2k-1 (see A006203, A046002, A046004, A046006. A046008, A046010, A046012, A046014, A046016 A046018, A046020). - T. D. Noe, May 07 2008

			a(15)=2 because the forms x^2 + xy + 4y^2 and 2x^2 + xy + 2y^2 have discriminant -15.

See A106856.

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

Cf. A106856 (start of many quadratic forms).
Cf. A133675 (n such that a(n)=1).
Cf. A223708 (without zeros).

dLim=150; cnt=Table[0, {dLim}]; nn=Ceiling[dLim/4]; Do[d=b^2-4a*c; If[GCD[a, b, c]==1 && 0<-d<=dLim, cnt[[ -d]]++ ], {b, 0, nn}, {a, b, nn}, {c, a, nn}]; cnt

{a(n)=local(m); if(n<3, 0, forvec(v=vector(3,k,[0,(n+1)\4]), if( (gcd(v)==1)&(-v[1]^2+4*v[2]*v[3]==n), m++ ), 1); m)} /* Michael Somos, May 31 2005 */

A267195 The j-invariants in A032354 are perfect cubes, except for two terms that have an extra factor of 2 or 3. Ignore these two extra factors and take the cube roots.

Original entry on oeis.org

0, 12, -15, 20, -32, 30, 66, -96, -160, 255, -960, -5280, -640320

Offset: 1

N. J. A. Sloane, Jan 27 2016

H. Cohn, Introduction to the Construction of Class Fields, Cambridge; p. 183.

Cf. A032354, A133675.

A329182 Negative discriminants with form class group isomorphic to C_2 X C_2 (negated).

Original entry on oeis.org

84, 96, 120, 132, 160, 168, 180, 192, 195, 228, 240, 280, 288, 312, 315, 340, 352, 372, 408, 435, 448, 483, 520, 532, 555, 595, 627, 708, 715, 760, 795, 928, 1012, 1435

Offset: 1

Jianing Song, Dec 05 2019

This sequence is finite and this is the full list.

Equivalently, negative discriminants of orders whose class group is isomorphic to C_2 X C_2 (negated). - Jianing Song, May 17 2021

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

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), A328825 (isomorphic to C_3), this sequence (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

The fundamental terms are listed in A192322. Cf. also A013658.

isA329182(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[2,2] \\ Jianing Song, May 17 2021

A330219 Negative discriminants with form class group isomorphic to C_4 (negated).

Original entry on oeis.org

39, 55, 56, 63, 68, 80, 128, 136, 144, 155, 156, 171, 184, 196, 203, 208, 219, 220, 252, 256, 259, 275, 291, 292, 323, 328, 355, 363, 387, 388, 400, 475, 507, 568, 592, 603, 667, 723, 763, 772, 955, 1003, 1027, 1227, 1243, 1387, 1411, 1467, 1507, 1555

Offset: 1

Jianing Song, Dec 05 2019

It seems that this is the full list.

Equivalently, negative discriminants of orders whose class group is isomorphic to C_4 (negated). - Jianing Song, May 17 2021

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

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), A328825 (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), this sequence (isomorphic to C_4).

Subsequence of A133676 and A317987. Cf. also A013658.

isA330219(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[4] \\ Jianing Song, May 17 2021

A225229 Numbers n such that if some integer N can be written in the form (a/2)^2+n(b/2)^2 for integers a and b, then every prime factor P of N which occurs to an odd power can also be written in the form (c/2)^2+n(d/2)^2 for integers c and d.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 11, 12, 19, 43, 67, 163

Offset: 1

V. Raman, Apr 30 2013

Alternately, values of n such that if for some number N, 4*N can be written as x^2+n*y^2, then for every prime factor p of N which occurs to an odd power, 4*p can be written as x^2+n*y^2.
For these values of n, for all primes p for which -n is a quadratic residue (mod p), 4*p can be written as x^2+n*y^2.
For n = 1, 2, 3 all primes p for which -n is a quadratic residue (mod p) can be written as x^2+n*y^2, with 4*p = (2*x)^2+n*(2*y)^2.
n = 1, 2, 3 are also the only values of n for which if some positive integer N can be written in the form a^2+n*b^2, for some integers a and b, then every prime factor P of N which occurs to an odd power can be written in the form c^2+n*d^2, for some integers c and d.
For n = 4, 7 all primes p for which -n is a quadratic residue (mod p) except for p = 2 can be written as x^2+n*y^2, with 4*p = (2*x)^2+n*(2*y)^2. For p = 2, 4 * 2 = 8 = 2^2 + 4*1^2 = 1^2 + 7*1^2.
For n = 8, all primes p (except for p = 2), for which -n is a quadratic residue (mod p) are generated by either x^2+8*y^2 or 3*x^2+2*x*y+3*y^2. We have 4*(x^2+8*y^2) = (2*x)^2+8*(2*y)^2 and 4*(3*x^2+2*x*y+3*y^2) = (2*x-2*y)^2+8*(x+y)^2. For p = 2, 4 * 2 = 8 = 0^2 + 8*1^2.
For n = 12, all primes p (except for p = 2), for which -n is a quadratic residue (mod p) are generated by either x^2+12*y^2 or 3*x^2+4*y^2. We have 4*(x^2+12*y^2) = (2*x)^2+12*(2*y)^2 and 4*(3*x^2+4*y^2) = (4*y)^2+12*x^2. We do not need to consider the prime p = 2 because numbers of form x^2+12*y^2 cannot contain prime factor of 2 raised to an odd power, as 12 is of the form 4^s*(8*t+3).
For n = 11, all primes p for which -n is a quadratic residue (mod p) are generated by either x^2+11*y^2 or 3*x^2+2*x*y+4*y^2. We have 4*(x^2+11*y^2) = (2*x)^2+11*(2*y)^2 and 4*(3*x^2+2*x*y+4*y^2) = (x+4*y)^2+11*x^2.
For n = 19, all primes p for which -n is a quadratic residue (mod p) are generated by either x^2+19*y^2 or 4*x^2+2*x*y+5*y^2. We have 4*(x^2+19*y^2) = (2*x)^2+19*(2*y)^2 and 4*(4*x^2+2*x*y+5*y^2) = (4*x+y)^2+19*y^2.
For n = 43, all primes p for which -n is a quadratic residue (mod p) are generated by either x^2+43*y^2 or 4*x^2+2*x*y+11*y^2. We have 4*(x^2+43*y^2) = (2*x)^2+43*(2*y)^2 and 4*(4*x^2+2*x*y+11*y^2) = (4*x+y)^2+43*y^2.
For n = 67, all primes p for which -n is a quadratic residue (mod p) are generated by either x^2+67*y^2 or 4*x^2+2*x*y+17*y^2. We have 4*(x^2+67*y^2) = (2*x)^2+67*(2*y)^2 and 4*(4*x^2+2*x*y+17*y^2) = (4*x+y)^2+67*y^2.
For n = 163, all primes p for which -n is a quadratic residue (mod p) are generated by either x^2+163*y^2 or 4*x^2+2*x*y+41*y^2. We have 4*(x^2+163*y^2) = (2*x)^2+163*(2*y)^2 and 4*(4*x^2+2*x*y+41*y^2) = (4*x+y)^2+163*y^2.

			n = 27 is not a member of this sequence because N = 27 = (0/2)^2 + 27*(2/2)^2 is of the form (x/2)^2+27*(y/2)^2, but for the prime factor 3 which appears to an odd power in N = 27, 3 is not of the form (x/2)^2+27*(y/2)^2 because 4 * 3 = 12 is not of the form x^2+27*y^2 (also 3 itself is not of the form x^2+27*y^2).

Cf. A133675, A003173 (squarefree values of this sequence).

cp's OEIS Frontend

A033195 Erroneous version of A133675.

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A032354 j-invariants for orders of class number 1.

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A133288 Negative discriminants with form class group of exponent 2 (negated).

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A322710 Negative discriminants with form class number 2 (negated).

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A328825 Negative discriminants with form class group isomorphic to C_3 (negated).

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A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

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A267195 The j-invariants in A032354 are perfect cubes, except for two terms that have an extra factor of 2 or 3. Ignore these two extra factors and take the cube roots.

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A329182 Negative discriminants with form class group isomorphic to C_2 X C_2 (negated).

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A330219 Negative discriminants with form class group isomorphic to C_4 (negated).

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A225229 Numbers n such that if some integer N can be written in the form (a/2)^2+n*(b/2)^2 for integers a and b, then every prime factor P of N which occurs to an odd power can also be written in the form (c/2)^2+n*(d/2)^2 for integers c and d.

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A225229 Numbers n such that if some integer N can be written in the form (a/2)^2+n(b/2)^2 for integers a and b, then every prime factor P of N which occurs to an odd power can also be written in the form (c/2)^2+n(d/2)^2 for integers c and d.