A278720 -id:A278720 - cp's OEIS frontend

A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).

1, -2, -3, 6, 2, 0, -1, -10, 0, -2, 10, 6, -7, 14, 0, -10, -12, 0, -6, 0, 9, -4, 10, 0, 18, -2, 0, 6, -14, -18, -11, 12, 0, 0, -22, 0, 20, 14, -6, 22, 0, 0, 23, -26, 0, -18, 4, 0, -14, -2, 0, -20, 0, 0, 0, 12, 3, 30, 26, 0, -30, 14, 0, 0, 2, 30, -28, -26, 0, -18, 10, 0, -13, -34, 0, 0, 20, 0, 26, 22, 0, -6, 0, 6, 18, 0

Offset: 0

N. J. A. Sloane

Number 49 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Glaisher (1884) essentially defines chi(n) as the sum over all solutions of n = x^2 + y^2 with even y and nonnegative odd x of x * (-1)^((x + y - 1)/2) and proves that it is multiplicative. If n is not == 1 (mod 4) then chi(n) = 0. - Michael Somos, Jun 18 2012
Denoted by g_2(q) in Cynk and Hulek on page 8 as the unique weight 2 level 32 newform. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012
The weight 2 level N = 32 newform (eta(q^4)*eta(q^8))^2 belongs to the elliptic curves y^2 = x^3 + 4*x , y^2 = x^3 - x, y^2 = x^3 - 11*x - 14 and y^2 = x^3 - 11*x + 14. See the Martin-Ono link, Theorem 2, row N = 32, and the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 26 2016

			G.f. = 1 - 2*x - 3*x^2 + 6*x^3 + 2*x^4 - x^6 - 10*x^7 - 2*x^9 + 10*x^10 + ...
G.f. (eta(q^4)*eta(q^8))^2 = q - 2*q^5 - 3*q^9 + 6*q^13 + 2*q^17 - q^25 - 10*q^29 - 2*q^37 + 10*q^41 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
J. E. Cremona, Algorithms for Modular Elliptic Curves.
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
T. Ishikawa, Congruences between binomial coefficients binomial(2f,f) and Fourier coefficients of certain eta-products, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to Glaisher's numbers

Cf. A000203, A002172, A278720, A279955.

A := Basis( ModularForms( Gamma0(32), 2), 341); A[2] - 2*A[6]; /* Michael Somos, Jun 12 2014 */

qEigenform( EllipticCurve( [0, 0, 0, -1, 0]), 341); /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(32), 2), 341)[1]; /* Michael Somos, Mar 25 2015 */

max=100; f[x_] := Product[(1-x^k)*(1-x^(2k)), {k, 1, max}]^2; CoefficientList[ Series[ f[x], {x, 0, max}], x](* Jean-François Alcover, Jan 04 2012, after g.f. *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, -1, 0], 1), 4*n + 1))}; /* Michael Somos, Jul 27 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^2, n))}; /* Michael Somos, Jul 27 2006 */

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor( 4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, (-p)^(e/2) * (1 + (-1)^e) / 2, forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0 = 1; y = a1 = x * (-1)^((x + y)\2) * if(y, 2, 1); for(i=2, e, x = y * a1 - p * a0; a0=a1; a1=x); a1 )))}; /* Michael Somos, Jun 18 2012 */

Expansion of (psi(x) * phi(-x))^2 = phi(-x) * f(-x^2)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q)^2 * eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [-2, -4, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) and b(p) = p - number of solutions of y^2 = x^3 - x (mod p). - Michael Somos, Jul 27 2006. b(p(n)) = A278720(n). - Wolfdieter Lang, Dec 26 2016
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^2.
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (Sum_{k>=0} (-1)^k * (2*k + 1) * x^(2*k + 1)^2) * (Sum_{k in Z} (-1)^k * x^(4*k)^2).
Coefficients of L-series for elliptic curve "32a2": y^2 = x^3 - x.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A276730 Number of solutions to y^2 == x^3 + 4*x (mod p) as p runs through the primes.

Original entry on oeis.org

2, 3, 7, 7, 11, 7, 15, 19, 23, 39, 31, 39, 31, 43, 47, 39, 59, 71, 67, 71, 79, 79, 83, 79, 79, 103, 103, 107, 103, 127, 127, 131, 159, 139, 135, 151, 135, 163, 167, 199, 179, 199, 191, 207, 199, 199, 211, 223, 227, 199, 207, 239, 271, 251, 255, 263, 295, 271, 295, 271

Offset: 1

Seiichi Manyama, Sep 16 2016

nonn

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(4t)*eta(8t))^2, see Theorem 2 in Martin & Ono.
It appears that a(n) = prime(n) iff prime(n) == 2 or 3 (mod 4). - Robert Israel, Sep 28 2016 This is true due to the L-function of this elliptic curve. See A278720. - Wolfdieter Lang, Dec 22 2016
The rational solutions of y^2 = x^3  + 4*x are (x,y) = (0,0), (2,4), (2,-4). See the Keith Conrad link, Corollary 3.17., p. 9. - Wolfdieter Lang, Dec 01 2016
For the p-defects p - N(p) see A278720. - Wolfdieter Lang, Dec 22 2016

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 == x^3 + 4*x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 1)
2,   3,       3:   (0, 0), (2, 1), (2, 2)
3,   5,       7:   (0, 0), (1, 0), (2, 1),
                   (2, 4), (3, 2), (3, 3),
                   (4, 0)
4,   7,       7:   (0, 0), (2, 3), (2, 4),
                   (3, 2), (3, 5), (6, 3),
                   (6, 4)
...
The solutions (x, y) of y^2 == x^3 - x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 0);
2,   3,       3:   (0, 0), (1, 0), (2, 0);
3,   5,       7:   (0, 0), (1, 0), (2, 1),
                   (2, 4), (3, 2), (3, 3),
                   (4, 0);
4,   7,       7:   (0, 0), (1, 0), (4, 2),
                   (4, 5), (5, 1), (5, 6),
                   (6, 0);
... - _Wolfdieter Lang_, Dec 22 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Keith Conrad, Expository papers, Proofs by Descent.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A095978, A272207, A278720.

seq(nops([msolve(y^2-x^3-4*x, ithprime(n))]),n=1..100); # Robert Israel, Sep 28 2016

require 'prime'
def A(a3, a2, a4, a6, n)
  ary = []
  Prime.take(n).each{|p|
    a = Array.new(p, 0)
    (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}
    ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}
  }
  ary
end
def A276730(n)
  A(0, 0, 4, 0, n)
end

a(n) is the number of solutions of the congruence y^2 == x^3 + 4*x (mod prime(n)), n >= 1.
a(n) is also the number
of solutions of the congruence y^2 == x^3 - x (mod prime(n)), n >= 1. - Wolfdieter Lang, Dec 22 2016 (See the Cremona link given in A278720).

A279392 Bisection of primes congruent to 1 modulo 4 (A002144), depending on the corresponding sum of the A002972 and 2*A002973 entries being congruent to 1 modulo 4 or not. Here we give the first case.

Original entry on oeis.org

13, 17, 41, 53, 89, 97, 109, 149, 157, 229, 233, 257, 281, 313, 317, 337, 353, 373, 397, 401, 421, 433, 457, 461, 557, 569, 577, 601, 641, 709, 733, 769, 797, 809, 829, 853, 857, 881, 953, 997, 1013, 1021, 1049, 1061, 1097, 1153, 1193, 1201, 1213, 1229, 1277, 1297

Offset: 1

Wolfdieter Lang, Dec 11 2016

The primes from A002144 (1 (mod 4) primes) have the property A002144(n) = A002972(n)^2 + (2*A002973(n))^2 = A(n)^2 + B(n)^2 with odd A(n) and even B(n). A bisection of A002144 is given depending on A(n) + B(n) == 1 (mod 4) (part I) or A(n) + B(n) == 3 (mod 4) (part II). The present sequence gives part I of this bisection. The other part II is given in A279393.

This bisection appears in the formula for the p-defects of the congruence y^2 == x^3 + 4*x (mod p) for primes p == 1 (mod 4). See A278720 where for nonvanishing entries the sign is conjectured to be + for these part I primes, and it is - for the part II primes from A279393.

			a(1) = 13 is the first prime from A002144 which has A + B = 1 (mod 4) because 13 = A002144(2) = A(2)^2 + B(2)^2 = 3^2 + (2*1)^2, and 3 + 2 = 5 == 1 (mod 4), and A002144(1) = 5 leads to A + B = 3 (mod 4), because 5 = 1^2 + (2*1)^2.

Cf. A002144, A002972, A002973, A278720, A279393.

A prime A002144(m) = A(m)^2 + B(m)^2 belongs to this sequence iff (-1)^((A(m)-1)/2 + B(m)/2) = +1, where A(m) = A002972(m) and B(m)/2 = A002973(m).

More terms from Jinyuan Wang, Apr 20 2025

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A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).

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A276730 Number of solutions to y^2 == x^3 + 4*x (mod p) as p runs through the primes.

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A279392 Bisection of primes congruent to 1 modulo 4 (A002144), depending on the corresponding sum of the A002972 and 2*A002973 entries being congruent to 1 modulo 4 or not. Here we give the first case.

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