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

1,5

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).

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A180507 A218889 A131213 * A105371 A038188 A310285

Adjacent sequences: A320257 A320258 A320259 * A320261 A320262 A320263

Keyword

nonn

Author

Zhi-Wei Sun, Oct 08 2018

Status

approved