cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A133675 Negative discriminants with form class number 1 (negated).

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163
Offset: 1

Views

Author

N. J. A. Sloane, May 16 2003

Keywords

Comments

The list on p. 260 of Cox is missing -12, the list in Theorem 7.30 on p. 149 is correct. - Andrew V. Sutherland, Sep 02 2012
Let b(k) be the number of integer solutions of f(x,y) = k, where f(x,y) is the principal binary quadratic form with discriminant d<0 (i.e., f(x,y) = x^2 - (d/4)*y^2 if 4|d, x^2 + x*y + ((1-d)/4)*y^2 otherwise), then this sequence lists |d| such that {b(k)/b(1): k>=1} is multiplicative. See Crossrefs for the actual sequences. - Jianing Song, Nov 20 2019

References

  • D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, pp. 149, 260.
  • D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Crossrefs

Cf. A014602, A003173.
The sequences {b(k): k>=0}: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A033716 (d=-12), A004531 (d=-16), A028641 (d=-19), A138805 (d=-27), A033719 (d=-28), A138811 (d=-43), A318984 (d=-67), A318985 (d=-163).
The sequences {b(k)/b(1): k>=1}: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A096936 (d=-12), A113406 (d=-16), A035171 (d=-19), A138806 (d=-27), A110399 (d=-28), A035147 (d=-43), A318982 (d=-67), A318983 (d=-163).

Programs

  • PARI
    ok(n)={(-n)%4<2 && quadclassunit(-n).no == 1} \\ Andrew Howroyd, Jul 20 2018

Extensions

Corrected by David Brink, Dec 29 2007

A110399 Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 4, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 5, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Michael Somos, Oct 22 2005

Keywords

Comments

Half the number of integer solutions to x^2 + 7*y^2 = n. - Jianing Song, Nov 20 2019

Examples

			G.f. = x + x^4 + x^7 + 2*x^8 + x^9 + 2*x^11 + 3*x^16 + 2*x^23 + ...

References

  • Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 302, Entry 17(ii).

Crossrefs

Cf. A033719 (number of integer solutions to x^2 + 7*y^2 = n).
Cf. A035162, A035182.
Similar sequences: A096936, A113406, A138806.

Programs

  • Mathematica
    f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], e + 1, (1 + (-1)^e)/2]; f[2, e_] := e - 1; f[7, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)
  • PARI
    {a(n) = my(x); if( n<1, 0, x = valuation(n, 2); abs(x -1) * sumdiv(n/2^x, d, kronecker(-28, d)))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, e-1,  p==7, 1, kronecker(-7, p)==-1, (1+(-1)^e)/2, e+1)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, A = x *O(x^n); polcoeff( (eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-2 * eta(x^7 + A)^-2 * eta(x^14 + A)^5 * eta(x^28 + A)^-2 - 1)/2, n))};

Formula

a(n) is multiplicative with a(2^e) = |e-1|, a(7^e)= 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).
G.f.: Sum_{k>0} Kronecker(-7, k) x^k/(1-(-x)^k).
G.f.: (theta_3(q)*theta_3(q^7) - 1)/2 where theta_3(q) = 1 + 2*(q + q^4 + q^9 + ...).
a(2*n + 1) = A035162(2*n + 1) = A035182(2*n + 1). A033719(n) = 2*a(n) if n > 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - Amiram Eldar, Nov 16 2023

A138805 Theta series of quadratic form x^2 + x*y + 7*y^2.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 4, 0, 6, 0, 0, 0, 4, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 6, 4, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 12, 2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Mar 30 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*q + 2*q^4 + 4*q^7 + 6*q^9 + 4*q^13 + 2*q^16 + 4*q^19 + 2*q^25 + ...

Links

Crossrefs

Cf. A004016, A138806, A248897.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(27), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[8] + 6*A[10] + 4*A[14] + 2*A[15]; /* Michael Somos, Sep 08 2015 */
  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -3, n/#] {1, 1, 0, 1, 1, 0, 1, 1, 3}[[Mod[#, 9, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^27] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^27], {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 14], n, 1)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^54 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^27 + A) * eta(x^108 + A))^2 + 4 * x^7 * (eta(x^4 + A) * eta(x^108 + A))^2 / (eta(x^2 + A) * eta(x^54 + A)), n))};
  • PARI
    {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, kronecker(-3, n/d) * [ 3, 1, 1, 0, 1, 1, 0, 1, 1][n%9 + 1]))}; /* Michael Somos, Sep 08 2015 */

Formula

Expansion of theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) in powers of q.
Expansion of phi(q) * phi(q^27) + 4 * q^7 * psi(q^2) * psi(q^54) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 27 sequence [ 2, -2, -2, 2, -2, 2, 2, -2, 6, 2, -2, -2, 2, -2, 2, 2, -2, -6, 2, -2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(3^e) = 3 if e>1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{i, j in Z} x^(i*i + i*j + 7*j*j).
a(3*n + 2) = a(4*n + 2) = 0.
a(n) = 2 * A138806(n) unless n=0. a(9*n) = A004016(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 29 2023
Showing 1-3 of 3 results.

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