A059886 -id:A059886 - cp's OEIS frontend

A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

4, 6, 12, 14, 20, 58, 12, 88, 112, 150, 60, 290, 12, 138, 732, 144, 124, 1088, 60, 670, 740, 570, 28, 13864, 360, 138, 3968, 1362, 252, 22058, 124, 320, 1972, 1146, 732, 10704, 124, 570, 12260, 15176, 124, 60470, 28, 11634, 195728, 282, 508, 116592, 2032

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) = number of orders of degree-n monic irreducible polynomials over GF(9).
Also, number of primitive factors of 9^n - 1. - Max Alekseyev, May 03 2022

Max Alekseyev, Table of n, a(n) for n = 1..690

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), this sequence (b=9), A059892 (b=10).
Cf. A000005, A008683, A027381, A053452, A057952, A058946, A274909.
Column k=9 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(9^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 9^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(9^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} mu(n/d)*tau(9^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 8, 13, 26, 5, 10, 16, 20, 40, 80, 11, 22, 121, 242, 7, 14, 28, 52, 56, 91, 104, 182, 364, 728, 1093, 2186, 32, 41, 82, 160, 164, 205, 328, 410, 656, 820, 1312, 1640, 3280, 6560, 757, 1514, 9841, 19682, 44, 61, 88, 122, 244, 484, 488, 671, 968, 1342

Offset: 1

Boris Putievskiy, May 29 2012

The elements m of row n, are also solutions to the equation: multiplicative order of 3 mod m = n, with gcd(m,3) = 1, cf. A053446.

			Triangle T(n,k) begins:
1,   2;
4,   8;
13, 26;
5,  10,  16,  20, 40, 80;
11, 22, 121, 242;
7,  14,  28,  52, 56, 91, 104, 182, 364, 728;

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 555-557.
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Alois P. Heinz, Rows n = 1..47, flattened (first 13 rows from Boris Putievskiy)
Eric Weisstein's World of Mathematics, Irreducible Polynomial
XIAO, Polynomial order (computes the order of an irreducible polynomial over a finite field GF(p))

Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
Column k=2 of A212737.
Column k=1 gives: A218356.

with(numtheory):
M:= proc(n) option remember;
      divisors(3^n-1) minus U(n-1)
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..15);  # Alois P. Heinz, Jun 02 2012

M[n_] := M[n] = Divisors[3^n - 1] ~Complement~ U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]]; Array[T, 15] // Flatten (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (3^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(3^d-1) = A059885(n).

A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 12, 24, 31, 62, 124, 13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624, 11, 22, 44, 71, 142, 284, 781, 1562, 3124, 7, 9, 14, 18, 21, 28, 36, 42, 56, 63, 72, 84, 93, 126, 168, 186, 217, 248, 252, 279, 372, 434, 504, 558, 651, 744, 868, 1116

Offset: 1

Boris Putievskiy, Jun 02 2012

The elements m of row n, are also solutions to the equation: multiplicative order of 5 mod m = n, with gcd(m,5) = 1, cf. A050977.

			Triangle T(n,k) begins:
   1,  2,   4;
   3,  6,   8, 12,  24;
  31, 62, 124;
  13, 16,  26, 39,  48,  52,  78,  104,  156, 208, 312, 624;
  11, 22,  44, 71, 142, 284, 781, 1562, 3124;
  ...

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 557-560.

Alois P. Heinz, Rows n = 1..32, flattened
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Cf. A212906, A059912, A058944, A059499, A059886-A059892.
Column k=3 of A212737.
Column k=1 gives: A218357.

with(numtheory):
M:= proc(n) option remember;
      `if`(n=1, {1, 2, 4}, divisors(5^n-1) minus U(n-1))
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..8);

M[n_] := M[n] = If[n == 1, {1, 2, 4}, Divisors[5^n-1] ~Complement~ U[n-1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]]; Array[T, 8] // Flatten (* Jean-François Alcover, Jun 10 2018, from Maple *)

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (5^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,4}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(5^d-1) = A059887.

A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 16, 24, 48, 9, 18, 19, 38, 57, 114, 171, 342, 5, 10, 15, 20, 25, 30, 32, 40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400, 2801, 5602, 8403, 16806, 36, 43, 72, 76, 86, 129, 144, 152, 172, 228, 258

Offset: 1

Boris Putievskiy, Jun 02 2012

The elements m of row n, are also solutions to the equation: multiplicative order of 7 mod m = n, with gcd(m,7) = 1, cf. A053450.

			Triangle T(n,k) begins:
  1,  2,  3,  6;
  4,  8, 12, 16, 24,  48;
  9, 18, 19, 38, 57, 114, 171, 342;
  5, 10, 15, 20, 25,  30,  32,  40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400;
  ...

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 560-562.
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Boris Putievskiy, Table of n, a(n) for n = 1..102
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Gang Xiao, (computes the order of an irreducible polynomial over a finite field GF(p))

Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
Column k=4 of A212737.
Column k=1 gives: A218358.

with(numtheory):
M:= proc(n) option remember;
      `if`(n=1, {1, 2, 3, 6}, divisors(7^n-1) minus U(n-1))
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..7);

M[n_] := M[n] = If[n == 1, {1, 2, 3, 6}, Divisors[7^n - 1] ~Complement~ U[n - 1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Sep 24 2022, from Maple code *)

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (7^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,3,6}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(7^d-1) = A059889(n).

cp's OEIS Frontend

A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

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A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.

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A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.

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Author

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Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula