A309012 - OEIS

%I #13 Jul 07 2019 02:22:48

%S 0,0,0,0,3,3,0,16,21,43,30,62,77,99,129,146,203,187,228,245,252,345,

%T 372,382,402,558,570,631,663,756,901,1114,961,1325,1398,1253,1571,

%U 1470,1601,1795,2024,1988,2349,2014,2184,2200,2728,3054,3084,3718,3386,3224,3018,3861,3866,4258,4361,4418,5110,4724

%N Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that (i^16 mod prime(n)) > (j^16 mod prime(n)).

%C Conjecture : Let p be an odd prime, and let N be the number of ordered pairs (i,j) with 0 < i < j < p/2 and (i^16 mod p) > (j^16 mod p). When p == 1 (mod 16), we have 2 | N. Also, N == |{0<k<p/4: Leg(k,p) = 1}| (mod 2) if p == 9 (mod 16), where Leg(k,p) denotes the Legendre symbol (k/p). When p == 3 or 5 (mod 8), we have N == floor[(p-3)/8] (mod 2).

%H Zhi-Wei Sun, <a href="/A309012/b309012.txt">Table of n, a(n) for n = 1..1000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1809.07766">Quadratic residues and related permutations and identities</a>, arXiv:1809.07766 [math.NT], 2018-2019.

%e a(5) = 3 with prime(5) = 11, and (2^16 mod 11) = 9 greater than (3^16 mod 11) = 3, (4^16 mod 11) = 4 and (5^16 mod 11)) = 5.

%t r[p_]:=r[p]=Sum[Boole[PowerMod[j,16,p]>PowerMod[k,16,p]],{k,2,p/2},{j,1,k-1}];

%t Print[Table[r[Prime[n]],{n,1,60}]]

%Y Cf. A000040, A010804, A319311, A319882, A319894, A319903, A320044.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Jul 06 2019