A058943 -id:A058943 - cp's OEIS frontend

A058950 Coefficients of monic primitive irreducible polynomials over GF(5) listed in lexicographic order.

12, 13, 112, 123, 133, 142, 1032, 1033, 1042, 1043, 1102, 1113, 1143, 1203, 1213, 1222, 1223, 1242, 1302, 1312, 1322, 1323, 1343, 1403, 1412, 1442, 10122, 10123, 10132, 10133, 10412, 10413, 10442, 10443, 11013, 11023, 11032, 11042, 11113

Offset: 1

N. J. A. Sloane, Jan 13 2001

T. D. Noe, Table of n, a(n) for n=1..354 (through degree 5)
R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209. Church's table extends through degree 5.

Cf. A058945.

Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.

car = 5; maxDegree = 5;
okQ[coefs_List] := Module[{P}, P = coefs.x^Range[Length[coefs] - 1, 0, -1]; coefs[[1]] == 1 && IrreduciblePolynomialQ[P, Modulus -> car] && PrimitivePolynomialQ[P, car]];
FromDigits /@ Select[Table[IntegerDigits[k, car], {k, car+1, car^(maxDegree + 1)}], okQ] (* Jean-François Alcover, Sep 10 2019 *)

More terms from Jean Gaumont (jeangaum87(AT)yahoo.com), Apr 16 2006

A058951 Coefficients of monic primitive irreducible polynomials over GF(7) listed in lexicographic order.

Original entry on oeis.org

12, 14, 113, 123, 125, 135, 145, 153, 155, 163, 1032, 1052, 1062, 1112, 1124, 1152, 1154, 1214, 1242, 1262, 1264, 1304, 1314, 1322, 1334, 1352, 1354, 1362, 1422, 1432, 1434, 1444, 1504, 1524, 1532, 1534, 1542, 1552, 1564, 1604, 1612, 1632, 1644, 1654

Offset: 1

N. J. A. Sloane, Jan 13 2001

T. D. Noe, Table of n, a(n) for n=1..206 (through degree 4)
R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209. Church's table extends through degree 3.

Cf. A058946.

Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.

car = 7; maxDegree = 4;
okQ[coefs_List] := Module[{P}, P = coefs.x^Range[Length[coefs] - 1, 0, -1]; coefs[[1]] == 1 && IrreduciblePolynomialQ[P, Modulus -> car] && PrimitivePolynomialQ[P, car]];
FromDigits /@ Select[Table[IntegerDigits[k, car], {k, car+1, car^(maxDegree + 1)}], okQ] (* Jean-François Alcover, Sep 10 2019 *)

More terms from Jean Gaumont (jeangaum87(AT)yahoo.com), Apr 16 2006

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

Vladeta Jovovic, Feb 09 2001

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

			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;
  ...

Alois P. Heinz, Rows n = 1..71, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
XIAO Gang, Polynomial order: computes the order of an irreducible polynomial over a finite field GF(p), WIMS.

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.

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

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 *)

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

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

A169887 Primes in carryless arithmetic mod 10.

Original entry on oeis.org

21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 403, 407, 421, 427, 443, 449, 463, 469, 481, 487, 551, 553, 557, 559, 603, 607, 623, 629, 641, 647, 661, 667, 683, 689, 801, 809, 821, 823, 847, 849, 867, 869, 881, 883

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

Define the units in carryless arithmetic mod 10 to be the numbers 1, 3, 7 and 9 (these divide any number). A prime is a number N, not a unit, whose only factorizations are of the form N = u * M, where u is a unit.

There are two types: e-type primes (A163396) and f-type (A169984).

			Examples of nonprimes: 2 = 2*51, 4 = 2*2, 10 = 52*85, 11 = 57*83, 101 = 13*17, 102 = 58 * 254 = 502 * 801, 103 = 53 * 251 = 507 * 809, 107 = 53 * 259 = 507 * 801, 108 = 58 * 256 = 502 * 809, 111 = 227 * 553.

N. J. A. Sloane, Table of n, a(n) for n = 1..253560 (All primes with <= 9 digits. Based on T. D. Noe's b-files for A058943 and A058955.)
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

A169887 = A163396 union A169984.
Cf. A004520, A059729, A168294, A168541, A169885, A169886, A169884, A169903 (primitive primes).
Cf. A169962.

A059478 Arrange irreducible polynomials over GF(2) in lexicographic order and write down the order of each polynomial.

Original entry on oeis.org

1, 1, 3, 7, 7, 15, 15, 5, 31, 31, 31, 31, 31, 31, 63, 9, 21, 63, 63, 63, 63, 63, 21, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 127, 51, 255, 255, 255, 17, 85, 255, 255, 255, 255, 255, 255, 85, 85, 255, 85, 255, 51, 85, 255

Offset: 1

Vladeta Jovovic, Feb 03 2001

nonn

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983.

Polynomial order: computes the order of an irreducible polynomial over a finite field F_p.

Cf. A058943-A058952.

A059913 Triangle T(n,k) of numbers of n degree irreducible polynomials over GF(2) which have order A059912(n,k), k=1..A059499(n).

Original entry on oeis.org

2, 1, 2, 1, 2, 6, 1, 2, 6, 18, 2, 4, 8, 16, 8, 48, 1, 2, 6, 30, 60, 2, 8, 176, 1, 2, 2, 2, 4, 6, 4, 6, 8, 12, 12, 24, 24, 36, 48, 144, 630, 3, 6, 18, 378, 756, 10, 12, 60, 300, 1800, 16, 32, 64, 128, 256, 512, 1024, 2048, 7710, 1, 1, 2, 6, 6, 6, 8, 12, 18, 24

Offset: 1

Vladeta Jovovic, Feb 09 2001

Row sums give A001037.

			There are 9 (cf. A001037) irreducible polynomials of degree 6 over GF(2): 1 of order 9, 2 of order 21 and 6 of order 63 (cf. A059912).
Triangle T(n,k) begins:
  2;
  1;
  2;
  1,  2;
  6;
  1,  2,   6;
  18;
  2,  4,   8, 16;
  8, 48;
  1,  2,   6, 30, 60;
  2,  8, 176;
  ...

Alois P. Heinz, Rows n = 1..71, flattened

Cf. A058943, A059478, A059499, A001037, A059912, A000010.

Prepend[Table[Map[EulerPhi[#]/n &, Complement[Divisors[2^n - 1],Union[Flatten[Table[Divisors[2^k - 1], {k, 1, n - 1}]]]]], {n, 2,20}], {2}] // Grid (* Geoffrey Critzer, Dec 02 2019 *)

T(n,k) = phi(A059912(n,k))/n, where phi = Euler totient function A000010.

A169903 Primitive primes in carryless arithmetic mod 10.

Original entry on oeis.org

21, 23, 25, 27, 29, 51, 56, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 551, 2023, 2027, 2043, 2047, 2061, 2069, 2081, 2089, 2207, 2209, 2221, 2223, 2263, 2267, 2281, 2287, 2401, 2407, 2421, 2423, 2441, 2449, 2483, 2489, 2603, 2609

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 11 2010

Define the units in carryless arithmetic mod 10 to be the numbers 1, 3, 7 and 9 (these divide any number). A prime is a number N, not a unit, whose only factorizations are of the form N = u * M, where u is a unit.
A prime is primitive if it is not the carryless product of a smaller prime and a unit.
A subsequence of A169887.

David Applegate, Table of n, a(n) for n = 1..843 (all primitive primes with <= 6 digits)
Index entries for sequences related to carryless arithmetic

Cf. A004520, A059729, A168294, A168541, A169885, A169886, A169884, A169887.

Cf. A058943, A058945.

A091254 Reducible polynomials over GF(2) in binary format.

Original entry on oeis.org

100, 101, 110, 1000, 1001, 1010, 1100, 1110, 1111, 10000, 10001, 10010, 10100, 10101, 10110, 10111, 11000, 11010, 11011, 11100, 11101, 11110, 100000, 100001, 100010, 100011, 100100, 100110, 100111, 101000, 101010, 101011, 101100

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

a(n) = A007088(A091242(n)). Cf. A058943.

A065020 Coefficients of irreducible polynomials over GF(3) listed in lexicographic order.

Original entry on oeis.org

2, 11, 12, 20, 101, 112, 122, 200, 1021, 1022, 1102, 1112, 1121, 1201, 1211, 1222, 2000, 10012, 10022, 10102, 10111, 10121, 10202, 11002, 11021, 11101, 11111, 11122, 11222, 12002, 12011, 12101, 12112, 12121, 12212, 20000, 100021, 100022

Offset: 1

Robert G. Wilson v, Nov 01 2001

nonn

Cf. A058943.

Do[a = Reverse[ IntegerDigits[n, gf]]; b = {0}; l = Length[a]; k = 1; While[k < l + 1, b = Append[b, a[[k]]*x^(k - 1)]; k++ ]; b = Apply[ Plus, b]; c = Factor[b, Modulus -> gf]; If[ !IntegerQ[ Log[ gf, n]] && b == c, Print[ FromDigits[ IntegerDigits[n, gf]]]], {n, 1, 300} ]

A071802 Table in which n-th row gives exponents (in decreasing order) of lexicographically earliest primitive irreducible polynomial of degree n over GF(2).

Original entry on oeis.org

1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 2, 0, 6, 1, 0, 7, 1, 0, 8, 4, 3, 1, 0, 9, 1, 0, 10, 3, 0, 11, 2, 0, 12, 3, 0, 13, 4, 3, 1, 0, 14, 5, 0, 15, 1, 0, 16, 5, 3, 1, 0, 17, 3, 0, 18, 3, 0, 19, 5, 2, 1, 0, 20, 3, 0, 21, 2, 0, 22, 1, 0, 23, 5, 0, 24, 4, 3, 1, 0, 25, 3, 0, 26, 4, 3, 1, 0, 27, 5, 2, 1, 0

Offset: 1

N. J. A. Sloane, Jun 24 2002

			x+1, x^2+x+1, x^3+x+1, x^4+x+1, x^5+x^2+1, ...

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
M. Olofsson, VLSI Aspects on Inversion in Finite Fields, Dissertation No. 731, Dept Elect. Engin., Linkoping, Sweden, 2002.

Cf. A058943.

a = {}; Do[k = 2^n + 1; While[s = Apply[Plus, IntegerDigits[k, 2]*x^Table[i, {i, n, 0, -1}]]; k < 2^(n + 1) - 1 && Factor[s, Modulus -> 2] =!= s, k += 2]; a = Append[a, Reverse[ Exponent[ Apply[ Plus, IntegerDigits[k, 2]*x^Table[i, {i, n, 0, -1}]], x, List]]], {n, 1, 27}]; Flatten[a]

Extended by Robert G. Wilson v, Jun 25 2002

cp's OEIS Frontend

A058950 Coefficients of monic primitive irreducible polynomials over GF(5) listed in lexicographic order.

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A058951 Coefficients of monic primitive irreducible polynomials over GF(7) listed in lexicographic order.

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

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A169887 Primes in carryless arithmetic mod 10.

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A059478 Arrange irreducible polynomials over GF(2) in lexicographic order and write down the order of each polynomial.

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A059913 Triangle T(n,k) of numbers of n degree irreducible polynomials over GF(2) which have order A059912(n,k), k=1..A059499(n).

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A169903 Primitive primes in carryless arithmetic mod 10.

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A091254 Reducible polynomials over GF(2) in binary format.

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A065020 Coefficients of irreducible polynomials over GF(3) listed in lexicographic order.

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A071802 Table in which n-th row gives exponents (in decreasing order) of lexicographically earliest primitive irreducible polynomial of degree n over GF(2).

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