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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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, 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
Sequence in context: A296491 A085653 A022462 * A241786 A019151 A367976
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Sep 30 2018
STATUS
approved

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Last modified March 1 06:29 EST 2024. Contains 370430 sequences. (Running on oeis4.)