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

2,4

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.

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.

Example

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).

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}]

Crossrefs

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

Sequence in context: A296491 A085653 A022462 * A241786 A019151 A367976

Adjacent sequences: A319891 A319892 A319893 * A319895 A319896 A319897

Keyword

nonn

Author

Zhi-Wei Sun, Sep 30 2018

Status

approved