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 1-4 of 4 results.

A028959 Theta series of quadratic form with Gram matrix [ 2, 1; 1, 12 ].

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 4, 0, 4, 2, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 2, 4, 2, 4, 4, 0, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 4, 0, 4, 0, 0, 0, 4, 4, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 4, 4, 0, 8, 0, 0
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

theta[2,1;1,2d](z)=theta_3(z)*theta_3((4d-1)z)+theta_2(z)*theta_2((4d-1)z), generalizing the formula for theta(A_2), which is the case d=1 - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 16 2000.
The number of integer solutions (x, y) to x^2 + x*y + 6*y^2 = n, discriminant -23. - Ray Chandler, Jul 12 2014
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*x + 2*x^4 + 4*x^6 + 4*x^8 + 2*x^9 + 4*x^12 + 2*x^16 + 4*x^18 + ...
G.f. = 1 + 2*q^2 + 2*q^8 + 4*q^12 + 4*q^16 + 2*q^18 + 4*q^24 + 2*q^32 + 4*q^36 + 2*q^46 + 4*q^48 + 2*q^50 + 4*q^52 + 4*q^54 + 4*q^64 + 6*q^72 + 4*q^78 + 8*q^96 + ...

References

  • Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See F_1, p. 195.

Links

Crossrefs

Cf. A028958.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^23] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^23], {x, 0, n}]; (* Michael Somos, Mar 28 2015 *)

Formula

Expansion of phi(x) * phi(x^23) + 4*x^6 * psi(x^2) * psi(x^46) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 28 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 28 2015
G.f.: (theta_3(z)*theta_3(23z) + theta_2(z)*theta_2(23z)).

A033217 Primes of form x^2 + 23*y^2.

Original entry on oeis.org

23, 59, 101, 167, 173, 211, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 829, 853, 877, 883, 991, 997, 1097, 1117, 1151, 1163, 1181, 1231, 1319, 1451, 1453, 1481, 1553, 1613, 1669, 1697, 1787, 1789, 1867, 1871, 1879, 1889, 1913, 2027, 2053, 2143, 2309, 2339, 2347, 2381, 2393, 2423, 2539, 2647, 2677, 2693, 2707, 2741, 2819
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Discriminant -23.
Also primes of the form x^2 + x*y + 6*y^2. - N. J. A. Sloane, Jun 02 2014
Also primes of the form x^2 - x*y + 6*y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that X^3-X+1 is split modulo p. E.g., X^3-X+1 = (X-33)*(X-40)*(X-94) modulo 167. - Julien Freslon (julien.freslon(AT)wanadoo.fr), Feb 24 2007
It appears that, if x > 0, then tau(p) = A000594(p) == 2 (mod 23). - Comment from Jud McCranie
In fact, this sequence appears to be the same as primes p such that RamanujanTau(p) == {1,2} (mod 23). - Ray Chandler, Dec 01 2016
Excluding the first term, this sequence is the intersection of A191021 and A256567. - Arkadiusz Wesolowski, Oct 03 2021
From Amiram Eldar, Jan 10 2025: (Start)
a(2)..a(10000) are the first terms of the sequence of primes p such that tau(p) == 2 (mod 23), where tau is Ramanujan's tau function (A000594).
Moree and Noubissie (2024) proved that the following 3 conditions for a prime p are equivalent:
1. tau(p) == 2 (mod 23).
2. p divides A000931(p+3) where A000931 is the Padovan sequence.
3. The number of distinct roots modulo p of the polynomial x^3 - x - 1 is 3. (End)

References

  • David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
  • Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992. See pp. 158-160, "Integer 23 - the Tau function".

Links

Crossrefs

Cf. A000594, A191021, A256567. Primes in A028958.

Programs

  • Mathematica
    QuadPrimes2[1, 0, 23, 10000] (* see A106856 *)
Join[{23}, nn=23; pMax=5000; Union[Reap[Do[p=x^2 + nn y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]]] (* Vincenzo Librandi, Sep 05 2016 *)
  • PARI
    isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ Arkadiusz Wesolowski, Oct 03 2021
  • PARI
    isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ Arkadiusz Wesolowski, Oct 09 2021

A028929 Numbers represented by quadratic form with Gram matrix [ 4, 1; 1, 6 ], divided by 2.

Original entry on oeis.org

0, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 29, 31, 32, 36, 39, 41, 46, 47, 48, 50, 52, 54, 58, 62, 64, 69, 71, 72, 73, 75, 78, 81, 82, 87, 92, 93, 94, 96, 98, 100, 104, 108, 116, 117, 118, 123, 124, 127, 128, 131, 138, 139, 141, 142, 144, 146, 147, 150, 151, 156
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Nonnegative integers of the form 2x^2 + x*y + 3y^2, a positive definite quadratic form of discriminant -23. - N. J. A. Sloane, Jul 09 2014

Links

Crossrefs

For primes see A106867. Cf. A028958, A033217.

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A309138 Nonnegative integers of the form x^2 + 23*y^2.

Original entry on oeis.org

0, 1, 4, 9, 16, 23, 24, 25, 27, 32, 36, 39, 48, 49, 59, 64, 72, 81, 87, 92, 93, 96, 100, 101, 104, 108, 117, 121, 123, 128, 141, 144, 156, 167, 169, 173, 192, 196, 207, 208, 211, 213, 216, 219, 223, 225, 232, 236, 243, 248, 256, 261, 271, 279, 288, 289, 307, 312
Offset: 1

Views

Author

Hugo Pfoertner, Jul 14 2019

Keywords

Comments

Discriminant of positive definite binary quadratic form: -92.

Links

Crossrefs

Primes in this sequence: A033217.
Cf. A028958.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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