A319882 Number of ordered pairs (i, j) with 0 < i < j < prime(n)/2 such that (i^4 mod prime(n)) > (j^4 mod prime(n)).

0, 0, 0, 3, 3, 10, 16, 21, 33, 54, 82, 85, 103, 125, 138, 165, 157, 204, 267, 259, 359, 422, 471, 504, 584, 564, 627, 713, 628, 1053, 960, 1213, 1017, 1278, 1275, 1367, 1522, 1671, 1661, 2118, 2038, 2005, 2242, 2330, 2234, 2418, 3194, 3112, 3126

Offset

2,4

Comments

Conjecture: Let p be any odd prime, and let t(p) be the number of ordered pairs (i,j) with 0 < i < j < p/2 and (i^4 mod p) > (j^4 mod p). If p is not congruent to 7 modulo 8, then t(p) == floor((p-1)/8) (mod 2). When p == 7 (mod 8), we have t(p) == (p+1)/8 + (h(-p)+1)/2 (mod 2), where h(-p) denotes the class number of the imaginary quadratic field Q(sqrt(-p)).

See also A319311, A319480 and A319894 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(5) = 3 since prime(5) = 11, and the only ordered pairs (i, j) with 0 < i < j < 11/2 and (i^4 mod 11) > (j^4 mod 11) are (2, 3), (2, 4), (3, 4).

Mathematica

f[k_, p_] := f[k, p] = PowerMod[k, 4, p]; 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, 50}]

Crossrefs

Cf. A000040, A000583, A319311, A319480, A319894.

Sequence in context: A330288 A262923 A367301 * A225860 A024313 A132331

Adjacent sequences: A319879 A319880 A319881 * A319883 A319884 A319885

Keyword

nonn

Author

Zhi-Wei Sun, Sep 30 2018

Status

approved