A035187 -id:A035187 - cp's OEIS frontend

A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -15.

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -15. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

QP = QPochhammer; s = (QP[q^3]*QP[q^5])^2/(QP[q]*QP[q^15])/q - 1/q + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-15, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

m = -15; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

{a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15,d)))} \\ Michael Somos, Aug 25 2006

{a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3||p==5, 1, if((p%15)!=2^valuation(p%15,2), (e+1)%2, (e+1))))))} \\ Michael Somos, Aug 25 2006

{a(n)=if(n<1, 0, (qfrep([2, 1;1, 8],n, 1)+qfrep([4, 1;1, 4], n, 1))[n])} \\ Michael Somos, Aug 25 2006

{a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^3+A)^2*eta(x^5+A)^2/eta(x+A)/eta(x^15+A), n))} \\ Michael Somos, Aug 25 2006

From Michael Somos, Aug 25 2006: (Start)
Expansion of -1 + (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...]. if a(0)=1.
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
Given g.f. A(x), then B(x) = 1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = -v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
G.f.: -1 + x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
G.f.: -1 + (1/2) * (Sum_{n,m} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
a(n) is multiplicative with a(3^e) = a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.
a(3*n) = a(n). a(n) = |A106406(n)| unless n=0. a(n) = A123864(n) unless n=0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15) = 1.622311... . - Amiram Eldar, Oct 11 2022

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 2, 0, 0, 2, 1, 0, 0, 0, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2, 1

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 21. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

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. A038893, A038894.

a[n_] := DivisorSum[n, KroneckerSymbol[21, #] &]; Array[a, 100] (* Amiram Eldar, Oct 11 2022 *)

my(m = 21); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(21, d)); \\ Amiram Eldar, Nov 19 2023

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((5+sqrt(21))/2)/sqrt(21) = 0.683807... . - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(21, d).
Multiplicative with a(p^e) = 1 if Kronecker(21, p) = 0 (p = 3 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(21, p) = -1 (p is in A038894), and a(p^e) = e+1 if Kronecker(21, p) = 1 (p is in A038893 \ {3, 7}). (End)

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

N. J. A. Sloane

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[6, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=6); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(6, d)); \\ Amiram Eldar, Nov 20 2023

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

N. J. A. Sloane

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[13, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=13); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(13, d)); \\ Amiram Eldar, Nov 18 2023

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

N. J. A. Sloane

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[17, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=17); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(17, d)); \\ Amiram Eldar, Nov 18 2023

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

N. J. A. Sloane

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[10, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=10); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(10, d)); \\ Amiram Eldar, Nov 18 2023

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

N. J. A. Sloane

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[12, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=12); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(12, d)); \\ Amiram Eldar, Nov 18 2023

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

N. J. A. Sloane

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

a[n_] := DivisorSum[n, KroneckerSymbol[28, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

my(m = 28); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(28, d)); \\ Amiram Eldar, Nov 19 2023

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

N. J. A. Sloane

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

a[n_] := DivisorSum[n, KroneckerSymbol[29, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

my(m = 29); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(29, d)); \\ Amiram Eldar, Nov 19 2023

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

N. J. A. Sloane

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

a[n_] := DivisorSum[n, KroneckerSymbol[37, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)

my(m = 37); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(37, d)); \\ Amiram Eldar, Nov 20 2023

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)

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A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -15.

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A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 21.

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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.

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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.

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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.

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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.

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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.

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A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -15.

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.

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.

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.

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.

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.

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.

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.

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.

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.