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A320260 Number of ordered pairs (j,k) with 0 < j < k < prime(n)/2 such that (j*(j+1) mod prime(n)) > (k*(k+1) mod prime(n)). 1

%I #6 Oct 08 2018 16:37:58

%S 0,0,1,1,3,8,13,10,19,41,44,70,83,75,100,143,167,210,188,225,290,306,

%T 322,401,503,554,481,541,634,686,848,858,1048,981,1203,1099,1468,1332,

%U 1421,1700,1646,1831,2054,2077,2135,2017,2356,2698,2712,2851,3022,3112,3386,3447,3838,3551,4062,3956,4466,4569

%N Number of ordered pairs (j,k) with 0 < j < k < prime(n)/2 such that (j*(j+1) mod prime(n)) > (k*(k+1) mod prime(n)).

%C Conjecture: Let p be a prime with p == 3 (mod 4), and let T(p) denote the number of ordered pairs (j,k) with 0 < j < k < p/2 and (j*(j+1) mod p) > (k*(k+1) mod p). Then T(p) == floor((p+1)/8) (mod 2).

%H Zhi-Wei Sun, <a href="/A320260/b320260.txt">Table of n, a(n) for n = 1..2000</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.

%e a(4) = 1 since prime(4) = 7 and (1*2 mod 7, 2*3 mod 7, 3*4 mod 7) = (1,6,5) with 6 > 5.

%t T[p_]:=T[p]=Sum[Boole[Mod[j(j+1),p]>Mod[k(k+1),p]],{k,2,(p-1)/2},{j,1,k-1}];Table[T[Prime[n]],{n,1,60}]

%Y Cf. A000040, A000217, A002378, A319311, A319480.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Oct 08 2018

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