A309012 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that (i^16 mod prime(n)) > (j^16 mod prime(n)).

0, 0, 0, 0, 3, 3, 0, 16, 21, 43, 30, 62, 77, 99, 129, 146, 203, 187, 228, 245, 252, 345, 372, 382, 402, 558, 570, 631, 663, 756, 901, 1114, 961, 1325, 1398, 1253, 1571, 1470, 1601, 1795, 2024, 1988, 2349, 2014, 2184, 2200, 2728, 3054, 3084, 3718, 3386, 3224, 3018, 3861, 3866, 4258, 4361, 4418, 5110, 4724

Offset

1,5

Comments

Conjecture : Let p be an odd prime, and let N be the number of ordered pairs (i,j) with 0 < i < j < p/2 and (i^16 mod p) > (j^16 mod p). When p == 1 (mod 16), we have 2 | N. Also, N == |{0<k<p/4: Leg(k,p) = 1}| (mod 2) if p == 9 (mod 16), where Leg(k,p) denotes the Legendre symbol (k/p). When p == 3 or 5 (mod 8), we have N == floor[(p-3)/8] (mod 2).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Zhi-Wei Sun, Quadratic residues and related permutations and identities, arXiv:1809.07766 [math.NT], 2018-2019.

Example

a(5) = 3 with prime(5) = 11, and (2^16 mod 11) = 9 greater than (3^16 mod 11) = 3, (4^16 mod 11) = 4 and (5^16 mod 11)) = 5.

Mathematica

r[p_]:=r[p]=Sum[Boole[PowerMod[j, 16, p]>PowerMod[k, 16, p]], {k, 2, p/2}, {j, 1, k-1}];
Print[Table[r[Prime[n]], {n, 1, 60}]]

Crossrefs

Cf. A000040, A010804, A319311, A319882, A319894, A319903, A320044.

Sequence in context: A100543 A039928 A247889 * A137259 A166553 A285863

Adjacent sequences: A309009 A309010 A309011 * A309013 A309014 A309015

Keyword

nonn

Author

Zhi-Wei Sun, Jul 06 2019

Status

approved