Andy Huchala - cp's OEIS frontend

User: Andy Huchala

Andy Huchala has authored 37 sequences. Here are the ten most recent ones:

A363147 Primes q == 1 (mod 4) such that there is at least one equivalence class of quaternary quadratic forms of discriminant q not representing 2.

193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 733, 757, 761, 769, 773, 797, 809, 821, 829, 853, 857, 877, 881, 929, 937

Offset: 1

Andy Huchala, May 18 2023

nonn

Andy Huchala, Table of n, a(n) for n = 1..20000
F. Hirzebruch, Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe, Annales scientifiques de l’É.N.S. 4e série, tome 11, no 1 (1978), p. 101-165. See page 135.
Jürg Kramer, On the linear independence of certain theta-series, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

Cf. A307250, A363148.

bound = 100
P = Primes()
p = 2
for i in range(bound):
    p = P.next(p)
    if p % 4 == 1:
        K1. = NumberField(x^2 - p)
        K2. = NumberField(x^2 + p)
        K3. = NumberField(x^2 + 3*p)
        zeta = K1.zeta_function()
        h2 = len(K2.class_group())
        h3 = len(K3.class_group())
        H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3))
        H = (H_plus+int((p + 19)/24))/2
        if H_plus-H>0:
            print(p)

A363148 a(n) gives the number of equivalence classes of quaternary quadratic forms of discriminant A363147(n) not representing 2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 4, 1, 2, 2, 1, 1, 4, 6, 2, 6, 5, 7, 1, 1, 7, 4, 2, 9, 10, 7, 13, 5, 8, 11, 3, 5, 15, 3, 5, 7, 6, 8, 14, 20, 3, 4, 17, 6, 9, 8, 15, 10, 19, 20, 26, 7, 20, 20, 12, 34, 7, 13, 32, 26, 10, 16, 16, 23, 11, 17, 41, 37, 11, 28, 46, 20, 28, 14, 17

Offset: 1

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Author

Andy Huchala, May 17 2023

Keywords

nonn

Comments

Conjecture: a(n) ~ c * A363147(n) ^ d where d is a constant which is roughly 1.51 and c is one of four constants, depending on the value of A363147(n) mod 24. See plots in files.

Examples

a(5) = 1 as there is only one equivalence class of quaternary quadratic form of discriminant A363147(5) = 277 not representing 2 (see A307250).

Links

Andy Huchala, Table of n, a(n) for n = 1..20000

Andy Huchala, Growth of A363147(n) vs a(n)

F. Hirzebruch, Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe, Annales scientifiques de l’É.N.S. 4e série, tome 11, no 1 (1978), p. 101-165. See page 135.

Jürg Kramer, On the linear independence of certain theta-series, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

Crossrefs

Cf. A307250, A363147.

Programs

Sage

bound = 100 P = Primes() p = 2 for i in range(bound): p = P.next(p) if p % 4 == 1: K1. = NumberField(x^2 - p) K2. = NumberField(x^2 + p) K3. = NumberField(x^2 + 3*p) zeta = K1.zeta_function() h2 = len(K2.class_group()) h3 = len(K3.class_group()) H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3)) H = (H_plus+int((p + 19)/24))/2 if H_plus-H>0: print(H_plus-H)

A362878 Theta series of 18-dimensional lattice Kappa_18.

Original entry on oeis.org

1, 0, 6480, 157680, 1596510, 9488016, 40681440, 140492880, 406046520, 1047312720, 2426695200, 5208293520, 10421250750, 19873356480, 35716191840, 62355291696, 104234541390, 169488573120, 267064691760, 413777075760, 619573504896, 920235334320, 1331744781600

Offset: 0

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Author

Andy Huchala, May 08 2023

Keywords

nonn

Comments

Theta series is an element of the space of modular forms on Gamma_1(3) with Kronecker character -3, weight 9, and dimension 4 over the integers.

Examples

G.f. = 1 + 6480*q^4 + 157680*q^6 + ...

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Andy Huchala, Table of n, a(n) for n = 0..20000

G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362877, A362879, A362880.

Programs

Magma

prec := 20; ls := [4,2,4,0,-2,4,0,-2,0,4,0,0,-2,0,4,-2,-2,0,0,0,4,-2,-1,1,0,0,0,4,-2,-1,0,-1,1,2,2,4,-2,-2,0,1,1,2,2,2,4,-2,0,-2,0,1,1,0,0,0,4,1,1,0,0,0,-2,0,-1,-1,-2,4,-2,-1,0,0,0,1,1,1,1,1,-2,4,0,-1,1,1,0,-1,1,0,0,-1,1,-1,4,0,0,0,0,0,0,0,0,0,0,0,0,-2,4,0,-1,0,0,1,1,0,1,1,-1,0,0,1,-1,4,0,0,1,0,-1,0,1,0,0,0,-1,0,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,0,1,4]; S := SymmetricMatrix(ls); L := LatticeWithGram(S); T := ThetaSeries(L, 8); M := ThetaSeriesModularFormSpace(L); B := Basis(M, prec); Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..4]]);

A362880 Theta series of 20-dimensional lattice Kappa_20.

Original entry on oeis.org

1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478, 69157229760, 151652810580, 311116423500, 607158951120, 1127694969072, 2020055770530, 3478103852940, 5829999042420, 9467119804680, 15046034533560

Offset: 0

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Author

Andy Huchala, May 08 2023

Keywords

nonn

Comments

Theta series is an element of the space of modular forms on Gamma_0(9) of weight 10 and dimension 11 over the integers.

Examples

G.f. = 1 + 15390*q^4 + 575160*q^6 + ...

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Andy Huchala, Table of n, a(n) for n = 0..20000

G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362879.

Programs

Magma

prec := 40; ls := [4,2,4,0,-2,4,0,-2,0,4,0,0,-2,0,4,-2,-2,0,0,0,4,-2,-1,1,0,0,0,4,-2,-1,0,-1,1,2,2,4,-2,-2,0,1,1,2,2,2,4,-2,0,-2,0,1,1,0,0,0,4,1,1,0,0,0,-2,0,-1,-1,-2,4,-2,-1,0,0,0,1,1,1,1,1,-2,4,0,-1,1,1,0,-1,1,0,0,-1,1,-1,4,0,0,0,0,0,0,0,0,0,0,0,0,-2,4,0,-1,0,0,1,1,0,1,1,-1,0,0,1,-1,4,0,0,1,0,-1,0,1,0,0,0,-1,0,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,0,1,4,1,0,-1,1,1,0,-1,-1,0,0,0,0,0,0,1,-1,0,1,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,1,0,1,0,4]; S := SymmetricMatrix(ls); L := LatticeWithGram(S); M := ThetaSeriesModularFormSpace(L); B := Basis(M, prec); coeffs := [1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478]; Coefficients(&+[coeffs[i]*B[i] :i in [1..11]]);

A362879 Theta series of 19-dimensional lattice Kappa_19.

Original entry on oeis.org

1, 0, 9396, 284528, 3309660, 21996036, 103632480, 384538752, 1195104618, 3253783500, 7971340896, 17905302720, 37530681590, 74139276672, 139067432280, 250102136592, 433070833500, 724358442744, 1178016364548, 1866143480400, 2883345017508, 4367172766500

Offset: 0

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Author

Andy Huchala, May 08 2023

Keywords

nonn

Comments

Theta series is an element of the space of modular forms on Gamma_0(12) of weight 19/2 and dimension 19 over the integers.

Examples

G.f. = 1 + 9396*q^4 + 284528*q^6 + ...

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Andy Huchala, Table of n, a(n) for n = 0..20000

G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362880.

Programs

Magma

prec := 30; coeffs := [1, 0, 9396, 284528, 3309660, 21996036, 103632480, 384538752, 1195104618, 3253783500, 7971340896, 17905302720, 37530681590, 74139276672, 139067432280, 250102136592, 433070833500, 724358442744, 1178016364548]; ls := [4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, -1, 0, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, -1, 4, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, 4, 1, 0, -1, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 4]; S := SymmetricMatrix(ls); L := LatticeWithGram(S); M := ThetaSeriesModularFormSpace(L); B := Basis(M,prec); Coefficients(&+[coeffs[i]*B[i] :i in [1..19]]);

A362877 Theta series of 17-dimensional lattice Kappa_17.

Original entry on oeis.org

1, 0, 4266, 81792, 737862, 3809280, 15406210, 47505792, 133390290, 312588288, 711232812, 1408787328, 2789963820, 4931371008, 8870944884, 14417119872, 24144502662, 36878456832, 58393537998, 84926534016

Offset: 0

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Author

Andy Huchala, May 07 2023

Keywords

nonn

Comments

Theta series is an element of the space of modular forms on Gamma_1(48) with Kronecker character 12 in modulus 48, weight 17/2, and dimension 66 over the integers.

Examples

G.f. = 1 + 4266*q^4 + 81792*q^6 + ...

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362878, A362879, A362880.

Programs

Magma

prec := 10; S := SymmetricMatrix([4,2,4,0,-2,4,0,-2,0,4,0,0,-2,0,4,-2,-2,0,0,0,4,-2,-1,1,0,0,0,4,-2,-1,0,-1,1,2,2,4,-2,-2,0,1,1,2,2,2,4,-2,0,-2,0,1,1,0,0,0,4,1,1,0,0,0,-2,0,-1,-1,-2,4,-2,-1,0,0,0,1,1,1,1,1,-2,4,0,-1,1,1,0,-1,1,0,0,-1,1,-1,4,0,0,0,0,0,0,1,0,1,-1,1,-1,1,4,0,0,0,0,-1,1,-1,0,0,0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,0,1,-1,1,0,0,-1,4,0,0,0,0,0,0,0,1,0,-1,1,-1,1,0,1,-1,4]); L := LatticeWithGram(S); T := ThetaSeries(L, 2*prec); [Coefficients(T)[2*i-1] : i in [1..prec]];

A362876 Theta series of 16-dimensional lattice Kappa_16.

Original entry on oeis.org

1, 0, 2772, 42624, 335052, 1545984, 5698860, 16297344, 42785244, 94440960, 204094296, 385391232, 730053060, 1240934400, 2151268128, 3374469504, 5476016700, 8115545088, 12477938100, 17677480320, 26111897640, 35570481408, 50909418000, 67336722432, 93433877268

Offset: 0

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Author

Andy Huchala, May 07 2023

Keywords

nonn

Comments

Theta series is an element of the space of modular forms on Gamma_0(12) of weight 8 and dimension 17 over the integers.

Examples

G.f. = 1 + 2772*q^4 + 42624*q^6 + ...

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Andy Huchala, Table of n, a(n) for n = 0..20000

G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A029897, A047628, A362875, A362877, A362878, A362879, A362880.

Programs

Magma

prec := 40; S := SymmetricMatrix([4,2,4,0,-2,4,0,-2,0,4,0,0,-2,0,4,-2,-2,0,0,0,4,-2,-1,1,0,0,0,4,-2,-1,0,-1,1,2,2,4,-2,-2,0,1,1,2,2,2,4,-2,0,-2,0,1,1,0,0,0,4,1,1,0,0,0,-2,0,-1,-1,-2,4,-2,-1,0,0,0,1,1,1,1,1,-2,4,0,-1,1,1,0,-1,1,0,0,-1,1,-1,4,0,0,0,0,0,0,1,0,1,-1,1,-1,1,4,0,0,0,0,-1,1,-1,0,0,0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,0,1,-1,1,0,0,-1,4]); L := LatticeWithGram(S); T := ThetaSeries(L, 32); M := ThetaSeriesModularFormSpace(L); B := Basis(M,prec); Coefficients(&+[Coefficients(T)[2*i-1]*B[i] : i in [1..17]]);

A362875 Theta series of 15-dimensional lattice Kappa_15.

Original entry on oeis.org

1, 0, 1746, 21456, 147150, 607536, 2036334, 5410800, 13282866, 27563184, 56679732, 102040272, 184563384, 302221728, 504866340, 763016400, 1202127174, 1728479808, 2575653198, 3561176016, 5127122304, 6797385072, 9531403128, 12329627616, 16701654486, 21199654080

Offset: 0

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Author

Andy Huchala, May 07 2023

Keywords

nonn

Comments

Theta series is an element of the space of modular forms on Gamma_1(48) with Kronecker character 12 in modulus 48, weight 15/2, and dimension 58 over the integers.

Examples

G.f. = 1 + 1746*q^4 + 21456*q^6 + 147150*q^8 + ...

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Andy Huchala, Table of n, a(n) for n = 0..20000

G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A029897, A047628, A362876, A362877, A362878, A362879, A362880.

Programs

Magma

prec := 70; S := SymmetricMatrix([4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1, -1, 1, 4, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 0, 0, -1, 4]); ls := [1, 0, 1746, 21456, 147150, 607536, 2036334, 5410800, 13282866, 27563184, 56679732, 102040272, 184563384, 302221728, 504866340, 763016400, 1202127174, 1728479808, 2575653198, 3561176016, 5127122304, 6797385072, 9531403128, 12329627616, 16701654486, 21199654080, 28230179220, 34817427648, 45678519396, 55628679312, 71267532432, 85814825328, 108809427618, 128313065808, 161435864196, 188866349856, 233000967122, 271038881664, 332652360024, 380052936000, 464058384948, 528207272064, 634933480440, 719891109360, 862226645076, 963402396336, 1151630548200, 1283383148256, 1511712192624, 1682610190272, 1980149372586, 2173335020640, 2553938906832, 2802302452080, 3252053197962, 3565107859680, 4134281599332, 4478370612624]; L := LatticeWithGram(S); M := ThetaSeriesModularFormSpace(L); B := Basis(M,prec); Coefficients(&+[ls[i] * B[i] : i in [1..58]]);

A351674 Discriminants of imaginary quadratic fields with class number 36 (negated).

Original entry on oeis.org

959, 1055, 1295, 1599, 1727, 1967, 2199, 2504, 2516, 2895, 3055, 3495, 3656, 3711, 3716, 3896, 3956, 4164, 4255, 4280, 4388, 4472, 4615, 4619, 4623, 4664, 4772, 5007, 5048, 5055, 5063, 5156, 5240, 5291, 5316, 5343, 5455, 5636, 5732, 5767, 5960, 6015, 6055

Offset: 1

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Author

Andy Huchala, Mar 27 2022

Keywords

nonn
fini
full

Comments

Sequence contains 668 terms; largest is 217627.

The class groups associated to 255 of the above discriminants are isomorphic to C_36, 374 have a class group isomorphic to C_18 X C_2, 16 have a class group isomorphic to C_12 X C_3, and the remaining 23 have a class group isomorphic to C_6 X C_6.

Links

Andy Huchala, Table of n, a(n) for n = 1..668

Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.

Eric Weisstein's World of Mathematics, Class Number

Crossrefs

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

Programs

Sage

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)]; [-a[0] for a in ls if a[1] == 36]

A351675 Discriminants of imaginary quadratic fields with class number 37 (negated).

Original entry on oeis.org

1487, 2447, 3391, 5839, 6367, 8147, 9803, 10739, 12343, 12583, 12967, 14767, 15259, 16927, 18947, 19403, 20011, 20147, 21139, 21587, 22807, 23371, 23627, 26731, 28283, 28307, 31699, 31723, 36691, 37171, 37243, 38371, 39139, 39451, 40531, 41659, 42283, 42443

Offset: 1

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Author

Andy Huchala, Mar 27 2022

Keywords

nonn
fini
full

Comments

Sequence contains 85 terms; largest is 158923.

The class group of Q[sqrt(-d)] is isomorphic to C_37 for all d in this sequence.

Links

Andy Huchala, Table of n, a(n) for n = 1..85

Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.

Eric Weisstein's World of Mathematics, Class Number

Crossrefs

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

Programs

Sage

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)]; [-a[0] for a in ls if a[1] == 37]

cp's OEIS Frontend

User: Andy Huchala

A363147 Primes q == 1 (mod 4) such that there is at least one equivalence class of quaternary quadratic forms of discriminant q not representing 2.

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Sage

A363148 a(n) gives the number of equivalence classes of quaternary quadratic forms of discriminant A363147(n) not representing 2.

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Sage

A362878 Theta series of 18-dimensional lattice Kappa_18.

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Magma

A362880 Theta series of 20-dimensional lattice Kappa_20.

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Magma

A362879 Theta series of 19-dimensional lattice Kappa_19.

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Magma

A362877 Theta series of 17-dimensional lattice Kappa_17.

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Magma

A362876 Theta series of 16-dimensional lattice Kappa_16.

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Magma

A362875 Theta series of 15-dimensional lattice Kappa_15.

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