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

Author

Zhi-Wei Sun, Sep 20 2018

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A000290, A319311.

Programs

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}]