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 1-8 of 8 results.

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

Original entry on oeis.org

1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612, -370944, -577738, 401856, 1217160, 987136, -6905934, 2727432, 10661420, -7109760, -4219488, -12830688, 18643272, 21288960, -25499225, 13865712, -73279080, 24647168
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of the cusp form of weight 12 for the full modular group.
It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).
M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 (mod p) are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane, Mar 25 2007
Number 1 of the 74 eta-quotients listed in Table I of Martin (1996).
With Dedekind's eta function and the discriminant Delta one has eta(z)^24 = Delta(z)/(2*Pi)^12 = Sum_{m >= 1} tau(m)*q^m, with q = exp(2*Pi*i*z), and z in the complex upper half plane, where i is the imaginary unit. Delta is the eigenfunction of the Hecke operator T_n (n >= 1) with eigenvalue tau(n): T_n Delta = tau(n) Delta. From this the formula for tau(m)*tau(n) given below in the formula section follows. See, e.g., the Koecher-Krieg reference, Lemma and Satz, p. 212. Or the Apostol reference, eq. (3) on p. 114 and the first part of section 6.13 on p. 131. - Wolfdieter Lang, Jan 26 2016
For the functional equation satisfied by the Dirichlet series F(s), Re(s) > 7, of a(n) see the Hardy reference, p. 173, (10.9.4). It is (2*Pi)^(-s) * Gamma(s) * F(s) = (2*Pi)^(s-12) * Gamma(12-s) * F(12-s). This is attributed to J. R. Wilton, 1929, on p. 185. - Wolfdieter Lang, Feb 08 2017
Conjecture: |a(n)| with n > 1 can never be a perfect power. This has been verified for n up to 10^6. - Zhi-Wei Sun, Dec 18 2024
Conjecture: The numbers |a(n)| (n = 1,2,3,...) are distinct. This has been verified for the first 10^6 terms. - Zhi-Wei Sun, Dec 21 2024
Conjecture: |a(n)| > 2*n^4 for all n > 2. This has been verified for n = 3..10^6. - Zhi-Wei Sun, Dec 25 2024
Conjecture: a(m)^2 + a(n)^2 can never be a perfect power. This implies Lehmer's conjecture that a(n) is never zero. We have verified that there is no perfect power among a(m)^2 + a(n)^2 with m,n <= 1000 . - Zhi-Wei Sun, Dec 28 2024
Conjecture: The equation |a(m)a(n)| = x^k with m < n, k > 1 and x >= 0 has no solution. This has been verified for m < n <= 5000. - Zhi-Wei Sun, Dec 29 2024
For some conjectures motivated by additive combinatorics, one may consult the link to Question 485138 at MathOverflow. - Zhi-Wei Sun, Jan 25 2025

Examples

			G.f. = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + ...
35328 = (-24)*(-1472) = a(2)*a(4) = a(2*4) + 2^11*a(2*4/4) = 84480 + 2048*(-24) = 35328. See a comment on T_n Delta = tau(n) Delta above. - _Wolfdieter Lang_, Jan 21 2016

References

  • Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
  • Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, lecture X, pp. 161-185.
  • Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
  • Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.
  • Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
  • Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, p. 139.
  • M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
  • Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
  • Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
  • Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
  • Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, see p. 482.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
  • Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
  • Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.

Links

Crossrefs

Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).
For a(n) mod N for various values of N see A046694, A098108, A126812-...
Cf. A006352, A099059, A099060, A262339, A292781.
For primes p such that tau(p) == -1 (mod 23) see A106867.
Cf. A010816, A004009. A013973.
Cf. A126832(n) = a(n) mod 5.

Programs

  • Julia
    using Nemo
function DedekindEta(len, r)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(r, len, z)
    [coeff(e, j) for j in 0:len - 1] end
RamanujanTauList(len) = DedekindEta(len, 24)
RamanujanTauList(28) |> println # Peter Luschny, Mar 09 2018
  • Magma
    M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);
  • Magma
    Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */
  • Maple
    M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);
  • Mathematica
    CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* Dean Hickerson, Jan 03 2003 *)
max = 28; g[k_] := -BernoulliB[k]/(2k) + Sum[ DivisorSigma[k - 1, n - 1]*q^(n - 1), {n, 2, max + 1}]; CoefficientList[ Series[ 8000*g[4]^3 - 147*g[6]^2, {q, 0, max}], q] // Rest (* Jean-François Alcover, Oct 10 2012, from modular forms *)
RamanujanTau[Range[40]] (* The function RamanujanTau is now part of Mathematica's core language so there is no longer any need to load NumberTheory`Ramanujan` before using it *) (* Harvey P. Dale, Oct 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
a[ n_] := With[{t = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ DedekindEta[t]^24, {q, 0, n}], {q, 0, n}]]; (* Michael Somos, May 27 2014 *)
  • PARI
    {a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};
  • PARI
    {a(n) = if( n<1, 0, polcoeff( x * (sum( i=1, (sqrtint( 8*n - 7) + 1) \ 2,(-1)^i * (2*i - 1) * x^((i^2 - i)/2), O(x^n)))^8, n))};
  • PARI
    taup(p,e)={
    if(e==1,
        (65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
    ,
        my(t=taup(p,1));
        sum(j=0,e\2,
            (-1)^j*binomial(e-j,e-2*j)*p^(11*j)*t^(e-2*j)
        )
    )
};
a(n)=my(f=factor(n));prod(i=1,#f[,1],taup(f[i,1],f[i,2]));
\\ Charles R Greathouse IV, Apr 22 2013
  • PARI
    \\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):
a(n) = n^4*sigma(n) - 24*sum(k=1, n-1, (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k));
vector(33, n, a(n)) \\ Joerg Arndt, Sep 06 2015
  • PARI
    a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016
  • Python
    from sympy import divisor_sigma
def A000594(n): return n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) # Chai Wah Wu, Nov 08 2022
  • Ruby
    def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A000594(n)
  ary = [1]
  a = [0] + (1..n - 1).map{|i| s(i)}
  (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
  ary
end
p A000594(100) # Seiichi Manyama, Mar 26 2017
  • Ruby
    def A000594(n)
  ary = [0, 1]
  (2..n).each{|i|
    s, t, u = 0, 1, 0
    (1..n).each{|j|
      t += 9 * j
      u += j
      break if i <= u
      s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u]
    }
    ary << s / (i - 1)
  }
  ary[1..-1]
end
p A000594(100) # Seiichi Manyama, Nov 25 2017
  • Sage
    CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013
  • Sage
    list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016

Formula

G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 04 2011
abs(a(n)) = O(n^(11/2 + epsilon)), abs(a(p)) <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.
Zagier says: The proof of these formulas, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u + 48*v + 4096*w) - v^3. - Michael Somos, Jul 19 2004
G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - Michael Somos, Dec 09 2013
a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - Michael Somos, Apr 17 2015
a(n) = tau(n) (with tau(0) = 0): tau(m)*tau(n) = Sum_{d| gcd(m,n)} d^11*tau(m*n/d^2), for positive integers m and n. If gcd(m,n) = 1 this gives the multiplicativity of tau. See a comment above with the Koecher-Krieg reference, p. 212, eq. (5). - Wolfdieter Lang, Jan 21 2016
Dirichlet series as product: Sum_{n >= 1} a(n)/n^s = Product_{n >= 1} 1/(1 - a(prime(n))/prime(n)^s + prime(n)^(11-2*s)). See the Mordell link, eq. (2). - Wolfdieter Lang, May 06 2016. See also Hardy, p. 164, eqs. (10.3.1) and (10.3.8). - Wolfdieter Lang, Jan 27 2017
a(n) is multiplicative with a(prime(n)^k) = sqrt(prime(n)^(11))^k*S(k, a(n) / sqrt(prime(n)^(11))), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2, and A076847(n) = a(prime(n)). See A076847 for alpha multiplicativity and examples. - Wolfdieter Lang, May 17 2016. See also Hardy, p. 164, eq. (10.3.6) rewritten in terms of S. - Wolfdieter Lang, Jan 27 2017
G.f. eta(z)^24 (with q = exp(2*Pi*i*z)) also (E_4(q)^3 - E_6(q)^2) / 1728. See the Hardy reference, p. 166, eq. (10.5.3), with Q = E_4 and R = E_6, given in A004009 and A013973, respectively. - Wolfdieter Lang, Jan 30 2017
a(n) (mod 5) == A126832(n).
a(1) = 1, a(n) = -(24/(n-1))*Sum_{k=1..n-1} A000203(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 26 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Euler Transform of [-24, -24, -24, -24, ...]. - Simon Plouffe, Jun 21 2018
a(n) = n^4*sigma(n)-24*Sum_{k=1..n-1} (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k). [See Douglas Niebur link]. - Wesley Ivan Hurt, Jul 22 2025

A028930 Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 6 ].

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The number of integer solutions to n = 2x^2 + xy + 3y^2. - Michael Somos, Oct 18 2005
In Osburn and Sahu (2010) the g.f. A(q) is denoted by F(z) where q = exp(2 pi i z). - Michael Somos, Sep 25 2013

Examples

			For n=24 the solutions are [2,2], [3,-2], [3,1] and their negatives, so a(24)=6.
G.f. = 1 + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^6 + 2*x^8 + 2*x^9 + 4*x^12 + ...
G.f. = 1 + 2*q^4 + 2*q^6 + 2*q^8 + 2*q^12 + 2*q^16 + 2*q^18 + 4*q^24 + 2*q^26 + 4*q^32 + 4*q^36 + 6*q^48 + 2*q^52 + 2*q^54 + 2*q^58 + 2*q^62 + 4*q^64 + 6*q^72 + ...

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 Phi_1, p. 195.

Links

Crossrefs

Cf. A030199, A106867, A224530.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(23), 1), 116); A[1] + 2*A[3] +2*A[4] +2*A[5] +2*A[7] + 2*A[9] + 2*A[10]; /* Michael Somos, Aug 24 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^46] + EllipticTheta[ 2, 0, q^2] EllipticTheta[ 2, 0, q^46] + (1/2) EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(23/2)], {q, 0, n}]; (* Michael Somos, Sep 25 2013 *)
terms = 105; max = Sqrt[terms] // Ceiling; s = Sum[x^(2 i^2 + i*j + 3 j^2), {i, -max, max}, {j, -max, max}]; CoefficientList[s, x][[1 ;; terms]] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *)
  • PARI
    {a(n) = if( n<1, n==0, 2 * qfrep( [4, 1; 1, 6], n, 1)[n])}; /* Michael Somos, Oct 18 2005 */
  • PARI
    list(n)=concat(1,2*Vec(qfrep([4,1;1,6],n,1))) \\ Charles R Greathouse IV, Sep 25 2013

Formula

G.f.: Sum_{i,j in Z} x^(2*i*i + i*j + 3*j*j). (This is the definition.) - Michael Somos, Sep 25 2013
Expansion of phi(q^2) * phi(q^46) + 2*q^3 * psi(q) * psi(q^23) + 4*q^12 * psi(q^4) * psi(q^92) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 25 2013, corrected by Sean A. Irvine, Feb 13 2020
G.f. A(q) = f(t_2(q)) where f() is the g.f. for A224530 and t_2(q) = eta(q) * eta(q^23) / A(q). - Michael Somos, Sep 25 2013
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, Sep 25 2013

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

A028958 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 12 ] (divided by 2).

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 12, 16, 18, 23, 24, 25, 26, 27, 32, 36, 39, 48, 49, 52, 54, 58, 59, 62, 64, 72, 78, 81, 82, 87, 92, 93, 94, 96, 100, 101, 104, 108, 116, 117, 121, 123, 124, 128, 138, 141, 142, 144, 146, 150, 156, 162, 164, 167, 169, 173, 174, 184, 186, 188, 192
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Nonnegative integers of the form x^2 + x*y + 6*y^2, discriminant -23. - Ray Chandler, Jul 12 2014
The Gram matrix is positive-definite, therefore, if w := (1 + sqrt(-23)) / 2, then |x + w*y|^2 = x^2 + x*y + 6*y^2 > 0 for all integers x and y except x = y = 0. - Michael Somos, Mar 28 2015
The theta function of the lattice with basis [1, w] is the g.f. of A028959, therefore, A028959(n) is positive if and only if n is in this sequence. - Michael Somos, Mar 28 2015

Links

Crossrefs

For primes see A033217. Cf. A028929, A106867.
Cf. A028959.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A278578 a(n) = least value of k such that tau(k) = A000594(k) == n mod 23.

Original entry on oeis.org

4, 1, 59, 3481, 5959, 12117361, 351581, 344322938, 995153, 35509681, 1223853461, 117428054, 58714027, 2447706922, 71019362, 1990306, 172161469, 703162, 24234722, 11918, 6962, 118, 2
Offset: 0

Views

Author

N. J. A. Sloane, Nov 29 2016

Keywords

Comments

A finite sequence, containing a(0) through a(22) only.

Examples

			tau(4) = -1472 is the first term of A000594 that is a multiple of 23, so a(0) = 4.

References

  • Wilton, John Raymond. "Congruence properties of Ramanujan's function τ(n)." Proceedings of the London Mathematical Society 2.1 (1930): 1-10. Table I gives all 23 terms.

Links

Crossrefs

Cf. A000594, A106867.

Extensions

More terms from Ray Chandler, Dec 01 2016

A106869 Primes of the form x^2+xy+6y^2, with x and y nonnegative.

Original entry on oeis.org

59, 101, 167, 173, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 877, 883, 991, 997, 1097, 1117, 1151, 1181, 1231, 1319, 1451, 1453, 1553, 1613, 1669, 1697, 1787, 1789, 1871, 1889, 1913, 2027, 2053, 2143, 2339, 2347
Offset: 1

Views

Author

T. D. Noe, May 09 2005

Keywords

Comments

Discriminant=-23.

Links

Crossrefs

Cf. A028929, A106867.

Programs

  • Mathematica
    QuadPrimes2[1, 1, 6, 10000] (* see A106856 *)

A106868 Primes of the form 2x^2-xy+3y^2, with x and y nonnegative.

Original entry on oeis.org

2, 3, 29, 31, 47, 73, 131, 151, 163, 193, 197, 233, 239, 277, 349, 353, 397, 487, 491, 499, 509, 547, 577, 601, 647, 653, 683, 811, 857, 859, 863, 929, 947, 1013, 1021, 1039, 1093, 1283, 1289, 1291, 1297, 1301, 1327, 1361, 1429, 1499, 1511, 1531
Offset: 1

Views

Author

T. D. Noe, May 09 2005

Keywords

Comments

Discriminant=-23.

Links

Crossrefs

Cf. A028929, A106867.

Programs

  • Mathematica
    QuadPrimes2[2, -1, 3, 10000] (* see A106856 *)

A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907
Offset: 1

Views

Author

T. D. Noe, May 19 2005

Keywords

Comments

Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.

Examples

			For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).

References

  • Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
  • Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.

Links

Crossrefs

Cf. A106312 (possible negative discriminants of cubic polynomials), A014602 (negative discriminants having class number 1), A006203 (negative discriminants having class number 3).
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