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

%I A273651 #41 Nov 09 2022 07:56:37
%S A273651 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,
%T A273651 74,50,34,73,29,109,64,4,137,66,32,154,64,106,51,119,97,95,110,63,102,
%U A273651 169,28,166
%N A273651 a(n) = A000594(p) mod p, where p = prime(n).
%H A273651 Chai Wah Wu, <a href="/A273651/b273651.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1000 from Seiichi Manyama)
%F A273651 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
%t A273651 Mod[RamanujanTau@ #, #] & /@ Prime@ Range@ 80 (* _Michael De Vlieger_, May 27 2016 *)
%o A273651 (Ruby)
%o A273651 require 'prime'
%o A273651 def mul(f_ary, b_ary, m)
%o A273651   s1, s2 = f_ary.size, b_ary.size
%o A273651   ary = Array.new(s1 + s2 - 1, 0)
%o A273651   s10 = [s1 - 1, m].min
%o A273651   (0..s10).each{|i|
%o A273651     s20 = [s2 - 1, m - i].min
%o A273651     (0..s20).each{|j|
%o A273651       ary[i + j] += f_ary[i] * b_ary[j]
%o A273651     }
%o A273651   }
%o A273651   ary
%o A273651 end
%o A273651 def power(ary, n, m)
%o A273651   return [1] if n == 0
%o A273651   k = power(ary, n >> 1, m)
%o A273651   k = mul(k, k, m)
%o A273651   return k if n & 1 == 0
%o A273651   return mul(k, ary, m)
%o A273651 end
%o A273651 def A000594(n)
%o A273651   ary = Array.new(n + 1, 0)
%o A273651   i = 0
%o A273651   j, k = 2 * i + 1, i * (i + 1) / 2
%o A273651   while k <= n
%o A273651     i & 1 == 1? ary[k] = -j : ary[k] = j
%o A273651     i += 1
%o A273651     j, k = 2 * i + 1, i * (i + 1) / 2
%o A273651   end
%o A273651   power(ary, 8, n).unshift(0)[1..n]
%o A273651 end
%o A273651 def A273651(n)
%o A273651   p_ary = Prime.each.take(n)
%o A273651   t_ary = A000594(p_ary[-1])
%o A273651   p_ary.inject([]){|s, i| s << t_ary[i - 1] % i}
%o A273651 end
%o A273651 p A273651(n)
%o A273651 (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
%o A273651 (Python)
%o A273651 from sympy import prime, divisor_sigma
%o A273651 def A273651(n):
%o A273651     p = prime(n)
%o A273651     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
%Y A273651 Cf. A000594, A007659, A273650.
%K A273651 nonn
%O A273651 1,6
%A A273651 _Seiichi Manyama_, May 27 2016