A319894 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that R(i^4,prime(n)) > R(j^4,prime(n)), where R(k,p) (with p an odd prime) denotes the unique integer r among 0,...,(p-1)/2 with k congruent to r or -r modulo p.

0, 0, 0, 6, 3, 5, 10, 22, 51, 62, 58, 53, 100, 146, 194, 200, 185, 246, 242, 310, 374, 344, 422, 497, 540, 582, 652, 683, 768, 946, 916, 1011, 1180, 1294, 1108, 1387, 1592, 1656, 1829, 2050, 2048, 2386, 2365, 2186, 2184, 2770, 2902, 2890, 3296, 3292, 3754, 3063, 3562, 3650, 4184, 4391, 4164, 4506, 4812

Offset: 2

Author

Zhi-Wei Sun, Sep 30 2018

Keywords

• nonn

Comments

Conjecture: Let p be an odd prime, and let r(p) be the number of ordered pairs (i,j) with 0 < i < j < p/2 and R(i^4,p) > R(j^4,p), where R(k,p) denotes the unique integer r among 0,...,(p-1)/2 with k congruent to r or -r modulo p. Then r(p) is even if p == 3 (mod 4). Also, r(p) == (p-5)/8 (mod 2) if p == 5 (mod 8). When p == 1 (mod 8), r(p) is even if and only if 2 is a quartic residue modulo p.

See also A319311, A319480 and A319882 for similar conjectures.

Examples

			a(6) = 3 since prime(6) = 13, (R(1^4,13),R(2^4,13),...,R(6^4,13)) = (1,3,3,9,1,9), and (2,5), (3,5) and (4,5) are only pairs (i,j) with 0 < i < j < 13/2 and R(i^4,13) > R(j^4,13).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..1000
• Zhi-Wei Sun, Quadratic residues and related permutations, arXiv:1809.07766 [math.NT], 2018.

Crossrefs

Cf. A000040, A000583, A319311, A319480, A319882.

Programs

Mathematica

f[k_,p_]:=f[k,p]=Abs[Mod[PowerMod[k,4,p],p,-p/2]];Inv[p_]:=Inv[p]=Sum[Boole[f[i,p]>f[j,p]],{j,2,(p-1)/2},{i,1,j-1}];Table[Inv[Prime[n]],{n,2,60}]