A276649 -id:A276649 - cp's OEIS frontend

A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).

7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, 4127, 5087, 5431, 6911, 8887, 9127, 9791, 9887, 12391, 13151, 14407, 15551, 16607, 19543, 20399, 21031, 21319, 21839, 23039, 25391, 26399, 28087, 28463, 28711, 29287, 33223, 39551, 43103, 44879, 46271

Offset: 1

Alexander Adamchuk, Nov 13 2009

nonn

Apparently A167860 is a subset of primes of the form 8*k + 7 (A007522).
Every A167859(m) from m=(p-1)/2 to m=(p-1) is divisible by prime p belonging to A167860.
7^3 divides A167859(13) and 7^2 divides A167859(10)-A167859(13).
Every A167859(m) from m=(kp-1 - (p-1)/2) to m=(kp-1) is divisible by prime p from A167860.
Every A167859(m) from m=((p^2-1)/2) to m=(p^2-1) is divisible by prime p from A167860. For p=7 every A167859(m) from m=((p^3-1)/2) to m=(p^3-1) and from m=((p^4-1)/2) to m(p^4-1)is divisible by p^2.

R. J. Mathar, Table of n, a(n) for n = 1..57

Cf. A000984, A006134, A007522, A066796, A082590, A132310, A144635, A167713, A167859, A276649.

A167859 := proc(n)
    option remember;
    if n <= 1 then
        add( (binomial(2*k, k)/2^k)^2, k=0..n) ;
        4^n*% ;
    else
        4*(5*n^2 - 4*n + 1)*procname(n-1) - 16*(2*n - 1)^2*procname(n-2) ;
        %/n^2 ;
    end if;
end proc:
isA167860 := proc(p)
    local m ;
    for m from (p-1)/2 to p-1 do
        if modp(A167859(m),p) > 0 then
            return false;
        end if;
    end do:
    true ;
end proc:
A167860 := proc(n)
    option remember ;
    if n = 0 then
        2;
    else
        p := nextprime(procname(n-1)) ;
        while not isA167860(p) do
            p := nextprime(p) ;
        end do ;
        return p;
    end if;
end proc:
seq(A167860(n),n=1..10) ; # R. J. Mathar, Jan 22 2025

is(p) = if(isprime(p)&&p%2, my(m=Mod(1, p), s=m); for(k=1, p\2, s+=(m*=(2*k-1)/k)^2); !s, 0); \\ Jinyuan Wang, Jul 24 2022

More terms from Jinyuan Wang, Jul 24 2022

A276807 Number of solutions of the congruence y^2 == x^3 - x^2 - 4*x + 4 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 4, 7, 7, 7, 15, 15, 23, 31, 23, 23, 31, 47, 39, 47, 55, 55, 63, 71, 63, 63, 87, 87, 95, 95, 119, 87, 119, 111, 95, 135, 135, 143, 151, 135, 167, 159, 151, 143, 167, 167, 175, 191, 191, 215, 183, 231, 231, 215, 207, 223, 255, 223, 231, 255, 271, 279, 263, 303, 255

Offset: 1

Seiichi Manyama, Sep 17 2016

nonn

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, eta(2t)*eta(4t)*eta(6t)*eta(12t), see Theorem 2 in Martin & Ono.

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

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

Cf. A276649.

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 A276807(n)
  A(0, -1, -4, 4, n)
end

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

A276847 Expansion of eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q.

Original entry on oeis.org

1, 0, -1, 0, -2, 0, 0, 0, 1, 0, 4, 0, -2, 0, 2, 0, 2, 0, -4, 0, 0, 0, -8, 0, -1, 0, -1, 0, 6, 0, 8, 0, -4, 0, 0, 0, 6, 0, 2, 0, -6, 0, 4, 0, -2, 0, 0, 0, -7, 0, -2, 0, -2, 0, -8, 0, 4, 0, 4, 0, -2, 0, 0, 0, 4, 0, -4, 0, 8, 0, 8, 0, 10, 0, 1, 0, 0, 0, -8, 0, 1, 0

Offset: 1

Seiichi Manyama, Sep 22 2016

The bisection of this sequence containing all nonzero terms is A030188.

Multiplicative. See A030188 for formula. - Andrew Howroyd, Jul 31 2018

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

Cf. A030188, A271231, A276807, A276649.

CoefficientList[Series[QPochhammer[x^2] QPochhammer[x^4] QPochhammer[x^6] QPochhammer[x^12], {x, 0, 100}], x] (* Jan Mangaldan, Jan 04 2017 *)

a(4*n-3) = A271231(4*n-3), a(4*n-2) = 0, a(4*n-1) = -A271231(4*n-1), a(4*n) = 0.
G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(6*k)) * (1 - x^(12*k)).
a(2*n+1) = A030188(n). - Michel Marcus, Sep 25 2016
Euler transform of period 12 sequence [0, -1, 0, -2, 0, -2, 0, -2, 0, -1, 0, -4, ...]. - Georg Fischer, Nov 17 2022

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A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).

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A276807 Number of solutions of the congruence y^2 == x^3 - x^2 - 4*x + 4 (mod p) as p runs through the primes.

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A276847 Expansion of eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q.

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