A058947 -id:A058947 - 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

A000020 Number of primitive polynomials of degree n over GF(2) (version 2).

Original entry on oeis.org

2, 1, 2, 2, 6, 6, 18, 16, 48, 60, 176, 144, 630, 756, 1800, 2048, 7710, 7776, 27594, 24000, 84672, 120032, 356960, 276480, 1296000, 1719900, 4202496, 4741632, 18407808, 17820000, 69273666, 67108864, 211016256, 336849900, 929275200, 725594112, 3697909056

Offset: 1

N. J. A. Sloane

nonn

The initial 2 should really be a 1. See A011260 for official version.

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
T. L. Booth, An analytical representation of signals in sequential networks, pp. 301-3240 of Proceedings of the Symposium on Mathematical Theory of Automata. New York, N.Y., 1962. Microwave Research Institute Symposia Series, Vol. XII; Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn, N.Y. 1963 xix+640 pp. See p. 303.
W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes. MIT Press, Cambridge, MA, 2nd edition, 1972, p. 476.
M. P. Ristenblatt, Pseudo-Random Binary Coded Waveforms, pp. 274-314 of R. S. Berkowitz, editor, Modern Radar, Wiley, NY, 1965; see p. 296.

David W. Wilson, Table of n, a(n) for n = 1..400
R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.
S. V. Duzhin and D. V. Pasechnik, Groups acting on necklaces and sandpile groups, 2014. See p. 92. - _N. J. A. Sloane_, Jun 30 2014

Cf. A058947, A011260 (with initial term 1).

Table[If[n==1,2,EulerPhi[2^n-1]/n],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

a(n)=if(n==1,2,eulerphi(2^n-1)/n) \\ Hauke Worpel (thebigh(AT)outgun.com), Jun 10 2008

A173270 Partial sums of A001037, the number of degree-n irreducible polynomials over GF(2).

Original entry on oeis.org

1, 3, 4, 6, 9, 15, 24, 42, 72, 128, 227, 413, 748, 1378, 2539, 4721, 8801, 16511, 31043, 58637, 111014, 210872, 401429, 766151, 1465021, 2807197, 5387992, 10359000, 19945395, 38458185, 74248452, 143522118, 277737798, 538038784, 1043325199

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

Cf. A001037, A058943 and A102569 for initial terms of underlying sequence. See also A058947, A011260, A059966, A000031 (n-bead necklaces but may have period dividing n).

a(n) = Sum_{i=0..n} A001037(i).

A337442 Number of output sequences from the linear feedback shift register whose feedback polynomial coefficients (excluding the constant term) correspond to the binary representation of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 2, 4, 6, 2, 4, 4, 2, 6, 4, 4, 8, 4, 2, 6, 2, 4, 8, 2, 4, 8, 4, 2, 6, 2, 2, 8, 14, 2, 6, 4, 8, 8, 4, 6, 6, 8, 12, 4, 4, 2, 8, 6, 2, 12, 8, 2, 8, 8, 2, 4, 4, 2, 4, 12, 6, 4, 6, 10, 20, 2, 4, 8, 2, 12, 6, 2, 2, 6, 4, 8, 16, 8, 2, 8, 4, 4, 16, 2

Offset: 0

Michael Schwartz, Aug 27 2020

a(n) > 1 for n > 0.
It appears that every term after a(2) is even.
It appears that a(2^n) is greater than each preceding term and is greater than or equal to each term up to a(2^(n+1)).
If a(n) = 2, then the nonzero shift register sequence is an m-sequence.

			For n = 3 = 11 in binary, the polynomial is 1+x+x^2 and the 2 shift register sequences are {00..., 01101...}.
For n = 4 = 100 in binary, the polynomial is 1+x^3 and the 4 shift register sequences are {000..., 001001..., 011011..., 111...}.
For n = 6 = 110 in binary, the polynomial is 1+x^2+x^3 and the 2 shift register sequences are {000..., 0010111001...}.
For n = 10 = 1010 in binary, the polynomial is 1+x^2+x^4 and the 4 shift register sequences are {0000..., 0001010001..., 0011110011..., 0110110...}.
For n = 11 = 1011 in binary, the polynomial in 1+x+x^2+x^4. Using a Fibonacci LSFR, if the current state of the register is 0001, the next input bit is 0+0+1=1, and the next state is 0011. If the current state is 0100, the next input bit is 0+0+0=0, and the next state is 1000. The 4 shift register sequences are {0000..., 00011010001..., 00101110010..., 1111...}.

a(2^n) = A000031(n+1).
A011260 counts how many 2's are in the interval [2^(n-1),(2^n)-1].
a(n) = 2 if and only if 2n+1 is in A091250.
Cf. A100447, A001037, A000016, A000013 (definition 2), A000020, A058947.
Cf. A011655..A011751 for examples of binary m-sequences.

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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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A000020 Number of primitive polynomials of degree n over GF(2) (version 2).

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A173270 Partial sums of A001037, the number of degree-n irreducible polynomials over GF(2).

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A337442 Number of output sequences from the linear feedback shift register whose feedback polynomial coefficients (excluding the constant term) correspond to the binary representation of n.

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