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%I #41 Nov 09 2022 07:56:37
%S 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 74,50,34,73,29,109,64,4,137,66,32,154,64,106,51,119,97,95,110,63,102,
%U 169,28,166
%N a(n) = A000594(p) mod p, where p = prime(n).
%H 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 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 Mod[RamanujanTau@ #, #] & /@ Prime@ Range@ 80 (* _Michael De Vlieger_, May 27 2016 *)
%o (Ruby)
%o require 'prime'
%o def mul(f_ary, b_ary, m)
%o s1, s2 = f_ary.size, b_ary.size
%o ary = Array.new(s1 + s2 - 1, 0)
%o s10 = [s1 - 1, m].min
%o (0..s10).each{|i|
%o s20 = [s2 - 1, m - i].min
%o (0..s20).each{|j|
%o ary[i + j] += f_ary[i] * b_ary[j]
%o }
%o }
%o ary
%o end
%o def power(ary, n, m)
%o return [1] if n == 0
%o k = power(ary, n >> 1, m)
%o k = mul(k, k, m)
%o return k if n & 1 == 0
%o return mul(k, ary, m)
%o end
%o def A000594(n)
%o ary = Array.new(n + 1, 0)
%o i = 0
%o j, k = 2 * i + 1, i * (i + 1) / 2
%o while k <= n
%o i & 1 == 1? ary[k] = -j : ary[k] = j
%o i += 1
%o j, k = 2 * i + 1, i * (i + 1) / 2
%o end
%o power(ary, 8, n).unshift(0)[1..n]
%o end
%o def A273651(n)
%o p_ary = Prime.each.take(n)
%o t_ary = A000594(p_ary[-1])
%o p_ary.inject([]){|s, i| s << t_ary[i - 1] % i}
%o end
%o p A273651(n)
%o (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 (Python)
%o from sympy import prime, divisor_sigma
%o def A273651(n):
%o p = prime(n)
%o 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 Cf. A000594, A007659, A273650.
%K nonn
%O 1,6
%A _Seiichi Manyama_, May 27 2016
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