cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A213224 Minimal order A(n,k) of degree-n irreducible polynomials over GF(prime(k)); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 7, 1, 3, 13, 5, 1, 4, 31, 5, 31, 1, 3, 9, 13, 11, 9, 1, 7, 7, 5, 11, 7, 127, 1, 3, 9, 16, 2801, 7, 1093, 17, 1, 4, 307, 5, 25, 36, 19531, 32, 73, 1, 3, 27, 5, 30941, 9, 29, 32, 757, 11, 1, 3, 7, 16, 88741, 63, 43, 64, 19, 44, 23
Offset: 1

Views

Author

Alois P. Heinz, Jun 06 2012

Keywords

Comments

Maximal order of degree-n irreducible polynomials over GF(prime(k)) is prime(k)^n-1 and thus A(n,k) < prime(k)^n.

Examples

			A(4,1) = 5: The degree-4 irreducible polynomials p over GF(prime(1)) = GF(2) are 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. Their orders (i.e., the smallest integer e for which p divides x^e+1) are 5, 15, 15, and the minimal order is 5. (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2).
Square array A(n,k) begins:
    1,    1,     1,    1,   1,       1,        1,   1, ...
    3,    4,     3,    4,   3,       7,        3,   4, ...
    7,   13,    31,    9,   7,       9,      307,  27, ...
    5,    5,    13,    5,  16,       5,        5,  16, ...
   31,   11,    11, 2801,  25,   30941,    88741, 151, ...
    9,    7,     7,   36,   9,      63,        7,   7, ...
  127, 1093, 19531,   29,  43, 5229043, 25646167, 701, ...
   17,   32,    32,   64,  32,      32,      128,  17, ...

Links

Crossrefs

Columns k=1-10 give: A212953, A218356, A218357, A218358, A218359, A218360, A218361, A218362, A218363, A218364.
Columns k=1-10 are first columns of: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341.
Cf. A212737 (all orders).

Programs

  • Maple
    with(numtheory):
M:= proc(n, i) option remember;
      divisors(ithprime(i)^n-1) minus U(n-1, i)
    end:
U:= proc(n, i) option remember;
      `if`(n=0, {}, M(n, i) union U(n-1, i))
    end:
A:= (n, k)-> min(M(n, k)[]):
seq(seq(A(n, d+1-n), n=1..d), d=1..14);
  • Mathematica
    M[n_, i_] := M[n, i] = Divisors[Prime[i]^n - 1] ~Complement~ U[n-1, i]; U[n_, i_] := U[n, i] = If[n == 0, {}, M[n, i] ~Union~ U[n-1, i]]; A[n_, k_] := Min[M[n, k]]; Table[Table[A[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Formula

A(n,k) = min(M(n,k)) with M(n,k) = {d : d|(prime(k)^n-1)} \ U(n-1,k) and U(n,k) = M(n,k) union U(n-1,k) for n>0, U(0,k) = {}.

A059912 Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).

Original entry on oeis.org

1, 3, 7, 5, 15, 31, 9, 21, 63, 127, 17, 51, 85, 255, 73, 511, 11, 33, 93, 341, 1023, 23, 89, 2047, 13, 35, 39, 45, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 8191, 43, 129, 381, 5461, 16383, 151, 217, 1057, 4681, 32767, 257, 771, 1285, 3855
Offset: 1

Views

Author

Vladeta Jovovic, Feb 09 2001

Keywords

Comments

A permutation of the odd positive numbers; namely, order each odd number d by the multiplicative order of 2 modulo d (in case of a tie, smaller d go first). - Jeppe Stig Nielsen, Feb 13 2020

Examples

			There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
Triangle T(n,k) begins:
    1;
    3;
    7;
    5,   15;
   31;
    9,   21,  63;
  127;
   17,   51,  85, 255;
   73,  511;
   11,   33,  93, 341, 1023;
  ...

Links

Crossrefs

Cf. A058943, A059478, A059499, A001037, A059913.
Cf. A212906, A212485, A212486.
Column k=1 of A212737.
Column k=1 gives: A212953.
Last elements of rows give: A000225.
Cf. A108974.

Programs

  • Maple
    with(numtheory):
M:= proc(n) option remember;
      divisors(2^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..20);  # Alois P. Heinz, May 31 2012
  • Mathematica
    m[n_] := m[n] = Complement[ Divisors[2^n - 1], u[n - 1]]; u[0] = {}; u[n_] := u[n] = Union[ m[n], u[n - 1]]; t[n_, k_] := m[n][[k]]; Flatten[ Table[t[n, k], {n, 1, 16}, {k, 1, Length[ m[n] ]}]] (* Jean-François Alcover, Jun 14 2012, after Alois P. Heinz *)
  • PARI
    maxDegree=26;for(n=1,maxDegree,forstep(d=1,2^n,2,znorder(Mod(2,d))==n&&print1(d,", "))) \\ inefficient, Jeppe Stig Nielsen, Feb 13 2020

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(2^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}. - Alois P. Heinz, Jun 01 2012

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

Views

Author

Boris Putievskiy, May 29 2012

Keywords

Comments

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.

Examples

			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;

References

  • 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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Views

Author

Boris Putievskiy, Jun 02 2012

Keywords

Comments

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.

Examples

			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;
  ...

References

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

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

Views

Author

Boris Putievskiy, Jun 02 2012

Keywords

Comments

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.

Examples

			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;
  ...

References

  • 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

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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).

A218336 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(11) listed in ascending order.

Original entry on oeis.org

1, 2, 5, 10, 3, 4, 6, 8, 12, 15, 20, 24, 30, 40, 60, 120, 7, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330, 16, 48, 61, 80, 122, 183, 240, 244, 305, 366, 488, 610, 732, 915, 976, 1220, 1464, 1830, 2440, 2928, 3660, 4880, 7320, 14640, 25, 50, 3221, 6442
Offset: 1

Views

Author

Alois P. Heinz, Oct 26 2012

Keywords

Examples

			Triangle begins:
   1,  2,    5,   10;
   3,  4,    6,    8,    12,    15,    20,     24,  30,  40, ...
   7, 14,   19,   35,    38,    70,    95,    133, 190, 266, ...
  16, 48,   61,   80,   122,   183,   240,    244, 305, 366, ...
  25, 50, 3221, 6442, 16105, 32210, 80525, 161050;
  ...

Links

Crossrefs

Column k=5 of A212737.
Last elements of rows give: A024127.
Column k=1 gives: A218359.
Row lengths are A212957(n,11).

Programs

  • Maple
    with(numtheory):
M:= proc(n) M(n):= divisors(11^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);
  • Mathematica
    M[n_] := M[n] = Divisors[11^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(11^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218337 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(13) listed in ascending order.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 7, 8, 14, 21, 24, 28, 42, 56, 84, 168, 9, 18, 36, 61, 122, 183, 244, 366, 549, 732, 1098, 2196, 5, 10, 15, 16, 17, 20, 30, 34, 35, 40, 48, 51, 60, 68, 70, 80, 85, 102, 105, 112, 119, 120, 136, 140, 170, 204, 210, 238, 240, 255, 272, 280, 336
Offset: 1

Views

Author

Alois P. Heinz, Oct 26 2012

Keywords

Examples

			Triangle begins:
:     1,     2,     3,      4,      6,     12;
:     7,     8,    14,     21,     24,     28,  42,  56,  84, 168;
:     9,    18,    36,     61,    122,    183, 244, 366, 549, ...
:     5,    10,    15,     16,     17,     20,  30,  34,  35, ...
: 30941, 61882, 92823, 123764, 185646, 371292;

Links

Crossrefs

Column k=6 of A212737.
Column k=1 gives: A218360.
Row lengths are A212957(n,13).

Programs

  • Maple
    with(numtheory):
M:= proc(n) M(n):= divisors(13^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);
  • Mathematica
    M[n_] := Divisors[13^n-1] ~Complement~ U[n-1]; U[n_] := If[n == 0, {}, M[n] ~Union~  U[n-1]]; T[n_] := Sort[M[n]]; Table[T[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(13^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218338 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(17) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 24, 32, 36, 48, 72, 96, 144, 288, 307, 614, 1228, 2456, 4912, 5, 10, 15, 20, 29, 30, 40, 45, 58, 60, 64, 80, 87, 90, 116, 120, 145, 160, 174, 180, 192, 232, 240, 261, 290, 320, 348, 360, 435, 464, 480, 522, 576, 580, 696, 720
Offset: 1

Views

Author

Alois P. Heinz, Oct 26 2012

Keywords

Examples

			Triangle begins:
      1,      2,      4,      8,      16;
      3,      6,      9,     12,      18,  24, 32, 36, 48, 72, ...
    307,    614,   1228,   2456,    4912;
      5,     10,     15,     20,      29,  30, 40, 45, 58, 60, ...
  88741, 177482, 354964, 709928, 1419856;

Links

Crossrefs

Column k=7 of A212737.
Column k=1 gives: A218361.
Row lengths are A212957(n,17).

Programs

  • Maple
    with(numtheory):
M:= proc(n) M(n):= divisors(17^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);
  • Mathematica
    M[n_] := M[n] = Divisors[17^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(17^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218339 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(19) listed in ascending order.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 4, 5, 8, 10, 12, 15, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360, 27, 54, 127, 254, 381, 762, 1143, 2286, 3429, 6858, 16, 48, 80, 144, 181, 240, 362, 543, 720, 724, 905, 1086, 1448, 1629, 1810, 2172, 2715, 2896, 3258, 3620, 4344, 5430
Offset: 1

Views

Author

Alois P. Heinz, Oct 26 2012

Keywords

Examples

			Triangle begins:
    1,   2,   3,   6,   9,   18;
    4,   5,   8,  10,  12,   15,   20,   24,   30,   36,   40, ...
   27,  54, 127, 254, 381,  762, 1143, 2286, 3429, 6858;
   16,  48,  80, 144, 181,  240,  362,  543,  720,  724,  905, ...
  151, 302, 453, 906, 911, 1359, 1822, 2718, 2733, 5466, 8199, ...
  ...

Links

Crossrefs

Column k=8 of A212737.
Column k=1 gives: A218362.
Row lengths are A212957(n,19).

Programs

  • Maple
    with(numtheory):
M:= proc(n) M(n):= divisors(19^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);
  • Mathematica
    M[n_] := M[n] = Divisors[19^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(19^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218340 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(23) listed in ascending order.

Original entry on oeis.org

1, 2, 11, 22, 3, 4, 6, 8, 12, 16, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528, 7, 14, 77, 79, 154, 158, 553, 869, 1106, 1738, 6083, 12166, 5, 10, 15, 20, 30, 32, 40, 53, 55, 60, 80, 96, 106, 110, 120, 159, 160, 165, 212, 220, 240, 265, 318, 330, 352, 424, 440
Offset: 1

Views

Author

Alois P. Heinz, Oct 26 2012

Keywords

Examples

			Triangle begins:
       1,      2,      11,      22;
       3,      4,       6,       8,  12,  16,  24,  33,   44, ...
       7,     14,      77,      79, 154, 158, 553, 869, 1106, ...
       5,     10,      15,      20,  30,  32,  40,  53,   55, ...
  292561, 585122, 3218171, 6436342;
  ...

Links

Crossrefs

Column k=9 of A212737.
Column k=1 gives: A218363.
Row lengths are A212957(n,23).

Programs

  • Maple
    with(numtheory):
M:= proc(n) M(n):= divisors(23^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);
  • Mathematica
    M[n_] := M[n] = Divisors[23^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(23^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.
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