A035147 -id:A035147 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane, May 16 2003

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

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.

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).

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

Corrected by David Brink, Dec 29 2007

A138811 Theta series of quadratic form x^2 + xy + 11y^2.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 4, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 4, 0, 0, 0

Offset: 0

Michael Somos, Mar 31 2008, Apr 05 2008

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

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035147.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), this sequence (d=-43).

A := Basis( ModularForms( Gamma1(43), 1), 87); A[1] + 2*A[2] + 2*A[5] + 2*A[10] + 4*A[12] + 4*A[14] + 2*A[17] + 4*A[18]; /* Michael Somos, Sep 07 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -43, #] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^43] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^43], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-43, #]&]];
2 Table[a[n], {n, 1, 100}]] (* Vincenzo Librandi, Sep 07 2018 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))};

a(n)=if(n, sumdivmult(n,d,kronecker(-43,d))*2, 0) \\ Charles R Greathouse IV, Nov 23 2021

Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
Expansion of phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 43 sequence [2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(43^e) = 1, b(p^e) = e + 1 if Kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
G.f.: Sum_{i,j in Z} x^(i*i + i*j + 11*j*j).
a(n) = 2 * A035147(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(43) = 0.958176... . - Amiram Eldar, Nov 21 2023

A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

Original entry on oeis.org

4, 9, 11, 13, 17, 23, 25, 31, 41, 43, 47, 49, 53, 59, 67, 79, 83, 97, 101, 103, 107, 109, 127, 139, 167, 173, 181, 193, 197, 229, 239, 251, 269, 271, 281, 283, 293, 307, 311, 317, 337, 353, 359, 361, 367, 379, 397, 401, 431, 439, 443, 461, 479, 487, 509

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-43))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,43) >= 0 and the squares of primes such that (p,43) = -1, where (p,43) is the Legendre symbol.
For primes p such that (p,43) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-43))/2], namely (x + y*(1+sqrt(-43))/2) and (x + y*(1-sqrt(-43))/2), where (x,y) is a solution to x^2 + x*y + 11*y^2 = p; for p = 43, (sqrt(-43)) is the unique ideal with norm p; for primes p with (p,43) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-43))/2) = norm((1 - sqrt(-43))/2) = 11;
norm((3 + sqrt(-43))/2) = norm((3 - sqrt(-43))/2) = 13;
norm((5 + sqrt(-43))/2) = norm((5 - sqrt(-43))/2) = 17;
norm((7 + sqrt(-43))/2) = norm((7 - sqrt(-43))/2) = 23;
...
norm((19 + sqrt(-43))/2) = norm((19 - sqrt(-43))/2) = 101.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011591, A106891, A191031, A184902.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035147.
The total number of elements with norm n is given by A138811.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), this sequence (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341788(n) = my(disc=-43); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A318982 a(n) = Sum_{d|n} Kronecker(-67, d).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 06 2018

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -67.
Half of the number of integer solutions to x^2 + x*y + 17*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-67)] counted up to association.
Inverse Moebius transform of A011596.

			G.f. = x + x^4 + x^9 + x^16 + 2*x^17 + 2*x^19 + 2*x^23 + x^25 + 2*x^29 + x^36 + 2*x^37 + 2*x^47 + x^49 + 2*x^59 + x^64 + x^67 + 2*x^68 + 2*x^71 + 2*x^73 + 2*x^76 + ...

Jianing Song, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318984.
Moebius transform gives A011596.
Number of integral elements with norm n in Q[sqrt(d)] counted up to association: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A035171 (d=-19), A035147 (d=-43), this sequence (d=-67), A318983 (d=-163).

a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-67, #] &]];
Table[a[n], {n, 1, 110}] (* Vincenzo Librandi, Sep 10 2018 *)

PARI
```
a(n) = sumdiv(n, d, kronecker(-67, d))
```

a(n) is multiplicative with a(67^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-67, p) = -1, a(p^e) = e + 1 if Kronecker(-67, p) = 1.
G.f.: Sum_{k>0} Kronecker(-67, k) * x^k / (1 - x^k).
A318984(n) = 2 * a(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(67) = 0.383806... . - Amiram Eldar, Dec 16 2023

A318983 a(n) = Sum_{d|n} Kronecker(-163, d).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 06 2018

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -163.
Half of the number of integer solutions to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)] counted up to association.
Inverse Moebius transform of A011615.

			G.f. = x + x^4 + x^9 + x^16 + x^25 + x^36 + 2*x^41 + 2*x^43 + 2*x^47 + x^49 + 2*x^53 + 2*x^61 + x^64 + 2*x^71 + ...

Jianing Song, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318985.
Moebius transform gives A011615.
Number of integral elements with norm n in Q[sqrt(d)] counted up to association: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A035171 (d=-19), A035147 (d=-43), A318982 (d=-67), this sequence (d=-163).

a[n_] := DivisorSum[n, KroneckerSymbol[-163, #] &]; Array[a, 100] (* Amiram Eldar, Dec 16 2023 *)

PARI
```
a(n) = sumdiv(n, d, kronecker(-163, d))
```

a(n) is multiplicative with a(163^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, a(p^e) = e + 1 if Kronecker(-163, p) = 1.
G.f.: Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k).
A318985(n) = 2 * a(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(163) = 0.246068... . - Amiram Eldar, Dec 16 2023

A035233 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -43.

Original entry on oeis.org

1, 4, 9, 11, 13, 16, 17, 23, 25, 31, 36, 41, 43, 44, 47, 49, 52, 53, 59, 64, 67, 68, 79, 81, 83, 92, 97, 99, 100, 101, 103, 107, 109, 117, 121, 124, 127, 139, 143, 144, 153, 164, 167, 169, 172, 173, 176, 181, 187, 188, 193, 196, 197, 207, 208, 212, 221, 225

Offset: 1

N. J. A. Sloane

nonn

Also, positive numbers of the form x^2 + xy + 11y^2 (discriminant -43).

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A106891 (Primes of the form x^2 + xy + 11y^2).

Cf. A035147, A031363, A035256.

m=-43; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

More terms from Colin Barker, Jun 19 2014

cp's OEIS Frontend

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

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A138811 Theta series of quadratic form x^2 + x*y + 11*y^2.

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A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

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A318982 a(n) = Sum_{d|n} Kronecker(-67, d).

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A318983 a(n) = Sum_{d|n} Kronecker(-163, d).

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A035233 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= -43.

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A138811 Theta series of quadratic form x^2 + xy + 11y^2.

A035233 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -43.