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)).

0, 0, 1, 1, 3, 8, 13, 10, 19, 41, 44, 70, 83, 75, 100, 143, 167, 210, 188, 225, 290, 306, 322, 401, 503, 554, 481, 541, 634, 686, 848, 858, 1048, 981, 1203, 1099, 1468, 1332, 1421, 1700, 1646, 1831, 2054, 2077, 2135, 2017, 2356, 2698, 2712, 2851, 3022, 3112, 3386, 3447, 3838, 3551, 4062, 3956, 4466, 4569

Offset: 1

Author

Zhi-Wei Sun, Oct 08 2018

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
• Zhi-Wei Sun, Quadratic residues and related permutations and identities, arXiv:1809.07766 [math.NT], 2018.

Crossrefs

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

Programs

Mathematica

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}]