A273651 a(n) = A000594(p) mod p, where p = prime(n).

0, 0, 0, 0, 1, 8, 10, 7, 1, 24, 21, 31, 30, 31, 27, 29, 14, 49, 64, 19, 67, 37, 20, 56, 20, 74, 50, 34, 73, 29, 109, 64, 4, 137, 66, 32, 154, 64, 106, 51, 119, 97, 95, 110, 63, 102, 169, 28, 166

Offset: 1

Author

Seiichi Manyama, May 27 2016

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Seiichi Manyama)

Crossrefs

Cf. A000594, A007659, A273650.

Programs

Mathematica

Mod[RamanujanTau@ #, #] & /@ Prime@ Range@ 80 (* Michael De Vlieger, May 27 2016 *)

PARI

a(n,p=prime(n))=(65*sigma(p, 11)+691*sigma(p, 5)-691*252*sum(k=1, p-1, sigma(k, 5)*sigma(p-k, 5)))/756%p \\ Charles R Greathouse IV, Jun 07 2016

Python

from sympy import prime, divisor_sigma
def A273651(n):
    p = prime(n)
    return -1680*sum(pow(i,4,p)*divisor_sigma(i)*divisor_sigma(p-i) for i in range(1,p+1>>1)) % p # Chai Wah Wu, Nov 08 2022

Ruby

require 'prime'
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  s10 = [s1 - 1, m].min
  (0..s10).each{|i|
    s20 = [s2 - 1, m - i].min
    (0..s20).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary
end
def power(ary, n, m)
  return [1] if n == 0
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A000594(n)
  ary = Array.new(n + 1, 0)
  i = 0
  j, k = 2 * i + 1, i * (i + 1) / 2
  while k <= n
    i & 1 == 1? ary[k] = -j : ary[k] = j
    i += 1
    j, k = 2 * i + 1, i * (i + 1) / 2
  end
  power(ary, 8, n).unshift(0)[1..n]
end
def A273651(n)
  p_ary = Prime.each.take(n)
  t_ary = A000594(p_ary[-1])
  p_ary.inject([]){|s, i| s << t_ary[i - 1] % i}
end
p A273651(n)

Formula

for n > 1, a(n) = -1680*Sum_{i=1..(p-1)/2} i**4*sigma(i)*sigma(p-i) mod p where p = prime(n). - Chai Wah Wu, Nov 08 2022