Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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