A319480 -id:A319480 - cp's OEIS frontend

A319882 Number of ordered pairs (i, j) with 0 (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

Zhi-Wei Sun, Sep 30 2018

nonn

Conjecture: Let p be any odd prime, and let t(p) be the number of ordered pairs (i,j) with 0 (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.

			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).

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.

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

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}]

A319894 Number of ordered pairs (i,j) with 0 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.

Original entry on oeis.org

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

Zhi-Wei Sun, Sep 30 2018

nonn

Conjecture: Let p be an odd prime, and let r(p) be the number of ordered pairs (i,j) with 0 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.

			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).

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.

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

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}]

A319903 Number of ordered pairs (i,j) with 0 (j^8 mod prime(n)).

Original entry on oeis.org

0, 0, 1, 2, 7, 5, 10, 22, 45, 48, 68, 53, 104, 127, 146, 200, 203, 250, 288, 312, 387, 318, 450, 557, 536, 745, 664, 581, 722, 797, 986, 1011, 1082, 1474, 1294, 1317, 1608, 1684, 1893, 2096, 1898, 2297, 2333, 2090, 2467, 2652, 2836, 3352, 3698, 3326, 3380, 2981, 3778, 3902, 4165, 4743, 4350, 4652, 4240

Offset: 2

Zhi-Wei Sun, Oct 01 2018

nonn

Conjecture 1: 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^8 mod p) > (j^8 mod p). When p == 1 (mod 8), we have 2 | N if and only if 2 is a quartic residue modulo p. Also, N is even if p == 3 (mod 8). When p == 5 (mod 8), we have N == (p-5)/8 (mod 2). If p == 7 (mod 8) then N == (h(-p)+1)/2 (mod 2), where h(-p) is the class number of the imaginary quadratic field Q(sqrt(-p)).
Conjecture 2: Let p be an odd prime, and let N' be the number of ordered pairs (i,j) with 0 < i < j < p/2 and R(i^8,p) > R(j^8,p), where R(k,p) denotes the unique integer r among 0,...,(p-1)/2 with k congruent to r or -r modulo p. When p == 9 (mod 16), we have 2 | N' if and only if 2 is a quartic residue modulo p. Also, N' == floor((p+1)/8) (mod 2) if p is not congruent to 9 modulo 16.
See also A319311, A319480, A319882 and A319894 for similar conjectures.

			a(4) = 1 since prime(4) = 7, and (R(1^8,7),R(2^8,7),R(3^8,7)) = (1,3,2) with R(2^8,7) > R(3^8,7).
a(5) = 2 since prime(5) = 11, and (R(1^8,11),...,R(5^8,11)) = (1,3,5,2,4) with R(2^8,11) > R(4^8,11), R(3^8,11) > R(4^8,11) and R(3^8,11) > R(5^8,11).

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.

Cf. A000040, A001016, A319311, A319480, A319882, A319894.

f[k_,p_]:=f[k,p]=PowerMod[k,8,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,2,60}]

A320260 Number of ordered pairs (j,k) with 0 < j < k < prime(n)/2 such that (j(j+1) mod prime(n)) > (k(k+1) mod prime(n)).

Original entry on oeis.org

0, 0, 1, 1, 3, 8, 13, 10, 19, 41, 44, 70, 83, 75, 100, 143, 167, 210, 188, 225, 290, 306, 322, 401, 503, 554, 481, 541, 634, 686, 848, 858, 1048, 981, 1203, 1099, 1468, 1332, 1421, 1700, 1646, 1831, 2054, 2077, 2135, 2017, 2356, 2698, 2712, 2851, 3022, 3112, 3386, 3447, 3838, 3551, 4062, 3956, 4466, 4569

Offset: 1

Zhi-Wei Sun, Oct 08 2018

nonn

Conjecture: Let p be a prime with p == 3 (mod 4), and let T(p) denote the number of ordered pairs (j,k) with 0 < j < k < p/2 and (j*(j+1) mod p) > (k*(k+1) mod p). Then T(p) == floor((p+1)/8) (mod 2).

			a(4) = 1 since prime(4) = 7 and (1*2 mod 7, 2*3 mod 7, 3*4 mod 7) = (1,6,5) with 6 > 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Zhi-Wei Sun, Quadratic residues and related permutations and identities, arXiv:1809.07766 [math.NT], 2018.

Cf. A000040, A000217, A002378, A319311, A319480.

T[p_]:=T[p]=Sum[Boole[Mod[j(j+1),p]>Mod[k(k+1),p]],{k,2,(p-1)/2},{j,1,k-1}];Table[T[Prime[n]],{n,1,60}]

cp's OEIS Frontend

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)).

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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.

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A319903 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that (i^8 mod prime(n)) > (j^8 mod prime(n)).

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A320260 Number of ordered pairs (j,k) with 0 < j < k < prime(n)/2 such that (j*(j+1) mod prime(n)) > (k*(k+1) mod prime(n)).

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A320260 Number of ordered pairs (j,k) with 0 < j < k < prime(n)/2 such that (j(j+1) mod prime(n)) > (k(k+1) mod prime(n)).