A319480 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that R(i^2,prime(n)) > R(j^2,prime(n)), where R(k,p) (with p an odd prime and k an integer) denotes the unique integer r among 0,1,...,(p-1)/2 for which k is congruent to r or -r modulo p.

0, 0, 1, 3, 7, 10, 14, 19, 41, 42, 74, 79, 85, 100, 154, 163, 207, 224, 245, 309, 318, 342, 449, 536, 590, 553, 581, 715, 738, 856, 912, 1085, 1037, 1324, 1229, 1477, 1442, 1491, 1785, 1730, 1952, 1986, 2240, 2316, 2191, 2474, 2748, 2836, 3176

Offset

2,4

Comments

Conjecture: Let p be any odd prime and let N(p) be the number of ordered pairs (i,j) with 0 < i < j < p/2 and R(i^2,p) > R(j^2,p). Then N(p) == floor((p+1)/8) (mod 2).

See also A319311 for a similar conjecture.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 2..1000

Example

a(3) = 0 since prime(3) = 5 and R(1^2,5) = 1 = R(2^2,5).
a(4) = 1 since prime(4) = 7, R(1^2,7) = 1 < R(2^2,7) = 3, R(1^2,7) < R(3^2,7) = 2, and R(2^2,7) = 3 > R(3^2,7) = 2.

Mathematica

R[k_, p_]:=R[k, p]=Abs[Mod[k, p, -p/2]];
t[p_]:=t[p]=Sum[Boole[R[i^2, p]>R[j^2, p]], {j, 2, (p-1)/2}, {i, 1, j-1}]; Table[t[Prime[n]], {n, 2, 50}]

Crossrefs

Cf. A000040, A000290, A319311.

Sequence in context: A140487 A310189 A293788 * A310190 A307203 A347638

Adjacent sequences: A319477 A319478 A319479 * A319481 A319482 A319483

Keyword

nonn

Author

Zhi-Wei Sun, Sep 20 2018

Status

approved