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

Author

Zhi-Wei Sun, Jul 06 2019

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

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

Programs

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}]]

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