A272207 -id:A272207 - cp's OEIS frontend

A273163 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A272207(n).

0, -2, -1, 2, 0, 2, -6, -4, 6, 6, -4, 2, 6, -10, -6, -6, 12, 2, 2, -12, 2, 8, 6, -6, 2, 6, 14, -6, 2, -6, 2, 0, 18, -4, -6, 20, -22, -10, 18, -6, -12, -10, -12, 26, 18, 8, -16, -10, -6, 14, -6, -24, 14, 0, -6, -18, 18, 20, 26, 6

Offset: 1

Wolfdieter Lang, May 20 2016

The modularity pattern series is the expansion of the 39th modular cusp form of weight 2 and level N = 20, given in Table I of the Martin reference, i.e., eta^2(2*z)*eta^2(10*z) = Sum_{n >= 1} c(n)*q^n in powers of q = exp(2*Pi*i*z), with Im(z) > 0, where eta is the Dedekind function. c(prime(n)) = a(n), also for the bad primes 2 and 5 (the discriminant of this elliptic curve is -2^4*5^2). See also A030205 for the expansion in powers of q^2 (after deleting the factor q^(-1/2)): A030205((prime(n)-1)/2) = a(n), n >= 2.
See the comment on the Martin-Ono reference in A272207 which implies that eta^2(2*z) * eta^2(10*z) provides the modularity sequence for this elliptic curve.
The elliptic curve y^2 = x^3 + x^2 - x has the same p-defects.
For the above defined q-series coefficients c one seems to have for primes not 2 and 5 c(prime(n)^k) = (sqrt(prime(n)))^k*S(k, a(n)/sqrt(prime(n))) with Chebyshev's S polynomials (A049310), for n = 2, and n >= 4 and k >= 2. This corresponds to alpha-multiplicativity with alpha(x) = x (weight 2) except for primes 2 and 5, obtainable formally from Sum_{n>=1, without factors 2 or 5} c(n)/n^s = Product_{n=2,n>=4} 1/(1 - a(n)/prime(n)^s + prime(n)/prime(n)^{2*s}) (absolute convergence of the product seems to hold for Re(s) > 1 needed for the prime 3 factor). For alpha-multiplicativity see the Apostol reference, pp. 138-139 (exercise 6). c(2*k) = 0, c(2*k+1) = A030205(k), k >= 0. Thus c(2^k) = c(2)^k = 0, and it seems that c(5^k) = A030205((5^k-1)/2) = c(5)^k = (-1)^k, for k >= 1. Then one has multiplicativity of {c(n)} with the given c(p^k) formula.
For this multiplicity see the Michael Somos Oct 31 2005 comment on A030205, where c(n) is called b(n). - Wolfdieter Lang, May 24 2016
The Dirichlet sum could then be written as Sum_{n>=1} c(n)/n^s = 1/(1 - (-1)/5^s)* Product_{n=2,n>=4} 1/(1 - a(n)/prime(n)^s + prime(n)/prime(n)^{2*s}).

			{c(n)} multiplicativity test for n = 3^2*5^2 = 225: c(225) = A030205(112) = +1. c(3^2*5^2) = c(3^2)*c(5^2) = (3*S(2, a(2)/sqrt(3)))*(-1)^2 = ((-2)^2 - 3)*(+1) = +1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A030205, A049310, A272207.

a(n) = prime(n) - A272207(n), n >= 1, where A272207(n) is the number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)).

A276030 Primes p such that A272207(p) = p.

Original entry on oeis.org

2, 11, 131, 251, 491, 599, 1439, 3371, 5639, 5879, 6971, 7079, 8039, 8291, 9839, 10799, 11171, 12119, 14879, 16931, 17159, 18839, 23039, 23159, 25919, 50291, 53411, 53639, 59051, 69371, 74771, 74891, 75239, 81119, 81359, 117839, 119039, 126839, 130811, 131771

Offset: 1

Seiichi Manyama, Sep 10 2016

nonn

These terms are the primes prime(A273163(n)) for which A273163(n) = 0.
These terms are the primes for which A276491(p) == 0 (mod p).
These terms are the primes p = prime(n) for which A276664(n) = p.
These terms are the primes prime(A276695(n)) for which A276695(n) = 0.

			2 = A272207(1) = prime(1),
11 = A272207(5) = prime(5),
131 = A272207(32) = prime(32),
251 = A272207(54) = prime(54).

Cf. A272207, A273163, A276491, A276664, A276695.

A276491 Expansion of qProduct_{k>=1} (1-q^(2k))^2(1-q^(10k))^2.

Original entry on oeis.org

1, 0, -2, 0, -1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, -6, 0, -4, 0, -4, 0, 6, 0, 1, 0, 4, 0, 6, 0, -4, 0, 0, 0, -2, 0, 2, 0, -4, 0, 6, 0, -10, 0, -1, 0, -6, 0, -3, 0, 12, 0, -6, 0, 0, 0, 8, 0, 12, 0, 2, 0, 2, 0, -2, 0, 2, 0, -12, 0, -12, 0, 2, 0, -2, 0, 0, 0, 8, 0, -11, 0, 6, 0, 6, 0, -12, 0, -6, 0, 4, 0, 8, 0, 4, 0, 2, 0, 0, 0, 6, 0, 14, 0, 4, 0, -6, 0, 2, 0, -4, 0, -6, 0, -6, 0, 2, 0, -12, 0, -11, 0, -12, 0, -1, 0, 2, 0, 20, 0, 0, 0, -8, 0, -4

Offset: 1

Seiichi Manyama, Sep 10 2016

Multiplicative. See A030205 for formula. - Andrew Howroyd, Aug 05 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A030205, A272207, A276030.

QPochhammer[x^2]^2*QPochhammer[x^10]^2 + O[x]^100 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2019 *)

seq(n)={Vec((eta(x^2 + O(x*x^n)) * eta(x^10 + O(x*x^n)))^2)} \\ Andrew Howroyd, Aug 05 2018

a(2n-1) = A030205(n-1), a(2n) = 0 for n > 0.
G.f.: (eta(x^2) * eta(x^10))^2. - Andrew Howroyd, Aug 05 2018
Euler transform of period 10 sequence [0, -2, 0, -2, 0, -2, 0, -2, 0, -4, ...]. - Georg Fischer, Nov 17 2022

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).

cp's OEIS Frontend

A273163 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A272207(n).

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A276030 Primes p such that A272207(p) = p.

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A276491 Expansion of qProduct_{k>=1} (1-q^(2k))^2(1-q^(10k))^2.

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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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cp's OEIS Frontend

A273163 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A272207(n).

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A276030 Primes p such that A272207(p) = p.

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A276491 Expansion of q*Product_{k>=1} (1-q^(2*k))^2*(1-q^(10*k))^2.

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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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A276491 Expansion of qProduct_{k>=1} (1-q^(2k))^2(1-q^(10k))^2.