A273651 - OEIS

%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