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 1-3 of 3 results.

A212737 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k lists the orders of degree-d irreducible polynomials over GF(prime(k)); listing order for each column: ascending d, ascending value.

Original entry on oeis.org

1, 1, 3, 1, 2, 7, 1, 2, 4, 5, 1, 2, 4, 8, 15, 1, 2, 3, 3, 13, 31, 1, 2, 5, 6, 6, 26, 9, 1, 2, 3, 10, 4, 8, 5, 21, 1, 2, 4, 4, 3, 8, 12, 10, 63, 1, 2, 3, 8, 6, 4, 12, 24, 16, 127, 1, 2, 11, 6, 16, 12, 6, 16, 31, 20, 17, 1, 2, 4, 22, 9, 3, 7, 8, 24, 62, 40, 51
Offset: 1

Views

Author

Alois P. Heinz, Jun 02 2012

Keywords

Examples

			For k=1 the irreducible polynomials over GF(prime(1)) = GF(2) of degree 1-4 are: x, 1+x; 1+x+x^2; 1+x+x^3, 1+x^2+x^3; 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. The orders of these polynomials p (i.e., the smallest integer e for which p divides x^e+1) are 1; 3; 7; 5, 15. (Example: (1+x^3+x^4) * (1+x^3+x^4+x^6+x^8+x^9+x^10+x^11) == x^15+1 (mod 2)). Thus column k=1 begins: 1, 3, 7, 5, 15, ... .
Square array A(n,k) begins:
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    3,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
    7,  4,  4,  3,  5,  3,  4,  3, 11,  4, ...
    5,  8,  3,  6, 10,  4,  8,  6, 22,  7, ...
   15, 13,  6,  4,  3,  6, 16,  9,  3, 14, ...
   31, 26,  8,  8,  4, 12,  3, 18,  4, 28, ...
    9,  5, 12, 12,  6,  7,  6,  4,  6,  3, ...
   21, 10, 24, 16,  8,  8,  9,  5,  8,  5, ...
   63, 16, 31, 24, 12, 14, 12,  8, 12,  6, ...
  127, 20, 62, 48, 15, 21, 18, 10, 16,  8, ...

Links

Crossrefs

Columns k=1-10 give: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341.

Programs

Formula

Formulae for the column sequences are given in A059912, A212906, ... .

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) = {}.

A218364 Minimal order of degree-n irreducible polynomials over GF(29).

Original entry on oeis.org

1, 3, 13, 16, 732541, 9, 49, 32, 14437, 11, 23, 37, 521, 147, 181, 17, 3911, 19, 1386659, 176, 637, 69, 131327761273, 288, 151, 53, 52813, 784, 59, 99, 36767, 128, 299, 1973, 71, 304, 149, 16759, 169, 41, 83, 43, 173, 368, 2613097, 47, 283, 153, 197, 125, 103
Offset: 1

Views

Author

Alois P. Heinz, Oct 27 2012

Keywords

Comments

a(n) < 29^n.

Links

Crossrefs

Cf. A213224, A218341.

Programs

  • Maple
    with(numtheory):
M:= proc(n) M(n):= divisors(29^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..10);
  • Mathematica
    M[n_] := M[n] = Divisors[29^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 51}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

Formula

a(n) = min(M(n)) with M(n) = {d : d|(29^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.
a(n) = A218341(n,1) = A213224(n,10).
Showing 1-3 of 3 results.

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