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.

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A035188 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 6.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 2, 1, 3, 0, 1, 0, 2, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 1, 1, 3, 0, 0, 2, 1, 0, 0, 2, 2, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 1, 2, 0, 3, 2, 0, 0, 0, 2, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 24. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038877, A097934.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[6, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=6); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(6, d)); \\ Amiram Eldar, Nov 20 2023

Formula

From Amiram Eldar, Oct 17 2022: (Start)
a(n) = Sum_{d|n} Kronecker(6, d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(5+2*sqrt(6)) / sqrt(6) = 0.935881... . (End)
Multiplicative with a(p^e) = 1 if Kronecker(6, p) = 0 (p = 2 or 3), a(p^e) = (1+(-1)^e)/2 if Kronecker(6, p) = -1 (p is in A038877), and a(p^e) = e+1 if Kronecker(6, p) = 1 (p is in A097934). - Amiram Eldar, Nov 20 2023

A035195 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 13.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 13. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038883, A038884.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[13, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=13); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(13, d)); \\ Amiram Eldar, Nov 18 2023

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((3+sqrt(13))/2)/sqrt(13) = 0.662735... . - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(13, d).
Multiplicative with a(13^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(13, p) = -1 (p is in A038884), and a(p^e) = e+1 if Kronecker(13, p) = 1 (p is in A038883 \ {13}). (End)

A035199 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 17.

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 5, 1, 2, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 3, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 6, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 7, 0, 0, 2, 3, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 17. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038889, A038890.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[17, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=17); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(17, d)); \\ Amiram Eldar, Nov 18 2023

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(4+sqrt(17))/sqrt(17) = 1.016084... . - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(17, d).
Multiplicative with a(17^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(17, p) = -1 (p is in A038890), and a(p^e) = e+1 if Kronecker(17, p) = 1 (p is in A038889 \ {17}). (End)

A035192 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 10.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 40. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038880, A097955.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[10, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=10); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(10, d)); \\ Amiram Eldar, Nov 18 2023

Formula

From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(10, d).
Multiplicative with a(p^e) = 1 if Kronecker(10, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(10, p) = -1 (p is in A038880), and a(p^e) = e+1 if Kronecker(10, p) = 1 (p is in A097955).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(sqrt(10)+3)/sqrt(10) = 1.1500865228... . (End)

A035194 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 12.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 2, 1, 1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 1, 2, 2, 1, 0, 0, 2, 0, 0, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 12. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A003630, A097933, A196530.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[12, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=12); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(12, d)); \\ Amiram Eldar, Nov 18 2023

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2+sqrt(3))/sqrt(3) = 0.760345... (A196530). - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(12, d).
Multiplicative with a(p^e) = 1 if Kronecker(12, p) = 0 (p = 2 or 3), a(p^e) = (1+(-1)^e)/2 if Kronecker(12, p) = -1 (p is in A003630), and a(p^e) = e+1 if Kronecker(12, p) = 1 (p is in A097933). (End)

A035210 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 28.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 28. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A003632, A296934.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[28, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)
  • PARI
    my(m = 28); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(28, d)); \\ Amiram Eldar, Nov 19 2023

Formula

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(28, d).
Multiplicative with a(p^e) = 1 if Kronecker(28, p) = 0 (p = 2 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(28, p) = -1 (p is in A003632), and a(p^e) = e+1 if Kronecker(28, p) = 1 (p is in A296934).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(3*sqrt(7)+8)/sqrt(7) = 1.046454884756... . (End)

A035211 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 29.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 29. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038902, A191022.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[29, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)
  • PARI
    my(m = 29); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(29, d)); \\ Amiram Eldar, Nov 19 2023

Formula

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(29, d).
Multiplicative with a(29^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(29, p) = -1 (p is in A038902), and a(p^e) = e+1 if Kronecker(29, p) = 1 (p is in A191022).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((sqrt(29)+5)/2)/sqrt(29) = 0.611766289562... . (End)

A035219 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 37.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 2, 0, 3, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 4, 2, 0, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 2, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 0, 0, 0, 5
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 37. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038914, A191027.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[37, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)
  • PARI
    my(m = 37); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(37, d)); \\ Amiram Eldar, Nov 20 2023

Formula

From Amiram Eldar, Nov 20 2023: (Start)
a(n) = Sum_{d|n} Kronecker(37, d).
Multiplicative with a(37^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(37, p) = -1 (p is in A038914), and a(p^e) = e+1 if Kronecker(37, p) = 1 (p is in A191027).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(sqrt(37)+6)/sqrt(37) = 0.819292168725... . (End)

A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 33.

Original entry on oeis.org

1, 2, 1, 3, 0, 2, 0, 4, 1, 0, 1, 3, 0, 0, 0, 5, 2, 2, 0, 0, 0, 2, 0, 4, 1, 0, 1, 0, 2, 0, 2, 6, 1, 4, 0, 3, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 2, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4, 0, 7, 0, 2, 2, 6, 0, 0, 0, 4, 0, 4, 1, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 33. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038907, A038908.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[33, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)
  • PARI
    my(m = 33); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(33, d)); \\ Amiram Eldar, Nov 19 2023

Formula

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(33, d).
Multiplicative with a(p^e) = 1 if Kronecker(33, p) = 0 (p = 3 or 11), a(p^e) = (1+(-1)^e)/2 if Kronecker(33, p) = -1 (p is in A038908), and a(p^e) = e+1 if Kronecker(33, p) = 1 (p is in A038907 \ {3, 11}).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(4*sqrt(33)+23)/sqrt(33) = 1.332797188186... . (End)

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
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