A035171 -id:A035171 - cp's OEIS frontend

A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.

1, 2, 0, 0, 2, 4, 0, 4, 0, 2, 0, 4, 0, 0, 0, 0, 2, 4, 0, 2, 4, 0, 0, 4, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 4, 0, 6, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 4, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 2, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 6, 4, 0, 0, 0

Offset: 0

N. J. A. Sloane

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

The number of integer solutions (x, y) to x^2 + x*y + 5*y^2 = n, discriminant -19. - Ray Chandler, Jul 12 2014

			G.f. = 1 + 2*x + 2*x^4 + 4*x^5 + 4*x^7 + 2*x^9 + 4*x^11 + 2*x^16 + 4*x^17 + 2*x^19 + ...

Robert Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 409, Eq. (19).

G. C. Greubel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035171.

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), this sequence (d=-19), A138811 (d=-43).

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -19, #] &] 2]; (* Michael Somos, Jun 14 2012 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-19, d)) * 2)}; /* Michael Somos, Feb 27 2007 */

{a(n) = if( n<1, n==0, qfrep([2, 1;1, 10], n, 1)[n] * 2)}; /* Michael Somos, Feb 27 2007 */

Theta series of quadratic form with Gram matrix [ 2, 1; 1, 10 ].
Expansion of phi(q) * phi(q^19) + 4 * q^5 * psi(q^2)* psi(q^38) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 27 2007
Moebius transform is period 19 sequence [2, -2, -2, 2, 2, 2, 2, -2, 2, -2, 2, -2, -2, -2, -2, 2, 2, -2, 0, ...]. - Michael Somos, Feb 27 2007
a(n) = 2*b(n) where b(n) is multiplicative with a(0) = 1,  b(19^e) = 1, b(p^e) = e + 1 if Kronecker(-19, p) = 1, b(p^e) = (1 + (-1)^e)/2 if Kronecker(-19, p) = -1. - Michael Somos, Feb 27 2007
a(n) = 2 * A035171(n) unless n = 0. - Jianing Song, Sep 06 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(19) = 1.441461... . - Amiram Eldar, Dec 16 2023

A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

Original entry on oeis.org

4, 5, 7, 9, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149, 157, 163, 169, 191, 197, 199, 229, 233, 239, 251, 263, 271, 277, 283, 311, 313, 347, 349, 353, 359, 367, 389, 397, 419, 443, 457, 461, 463, 467, 479, 491, 499, 503, 541, 557, 571

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-19))/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,19) >= 0 and the squares of primes such that (p,19) = -1, where (p,19) is the Legendre symbol.
For primes p such that (p,19) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-19))/2], namely (x + y*(1+sqrt(-19))/2) and (x + y*(1-sqrt(-19))/2), where (x,y) is a solution to x^2 + x*y + 5*y^2 = p; for p = 19, (sqrt(-19)) is the unique ideal with norm p; for primes p with (p,19) = -1, (p) is the only ideal with norm p^2.

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

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

Cf. A011585, A106863, A191019, A191063.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171.
The total number of elements with norm n is given by A028641.
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*), this sequence (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341787(n) = my(disc=-19); (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

A035243 Positive numbers of the form x^2+xy+5y^2 (discriminant -19).

Original entry on oeis.org

1, 4, 5, 7, 9, 11, 16, 17, 19, 20, 23, 25, 28, 35, 36, 43, 44, 45, 47, 49, 55, 61, 63, 64, 68, 73, 76, 77, 80, 81, 83, 85, 92, 95, 99, 100, 101, 112, 115, 119, 121, 125, 131, 133, 137, 139, 140, 144, 149, 153, 157, 161, 163, 169, 171, 172, 175, 176, 180, 187, 188, 191, 196, 197, 199, 207, 209, 215, 220, 225, 229, 233, 235, 239, 244, 245, 251, 252, 253, 256

Offset: 1

N. J. A. Sloane

nonn

Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= -19 (A035171). [amended by Georg Fischer, Sep 03 2020]

Cf. A035171.

For primes see A106863.

m=-19; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Sep 03 2020

Edited by N. J. A. Sloane, Jun 01 2014

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A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.

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A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

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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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A035243 Positive numbers of the form x^2+xy+5y^2 (discriminant -19).

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