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.

Previous Showing 11-14 of 14 results.

A058949 Coefficients of monic primitive irreducible polynomials over GF(3) listed in lexicographic order.

Original entry on oeis.org

11, 112, 122, 1021, 1121, 1201, 1211, 10012, 10022, 11002, 11122, 11222, 12002, 12112, 12212, 100021, 100211, 101011, 101201, 101221, 102101, 102211, 110021, 110101, 110111, 111011, 111121, 111211, 112001, 112111, 112201, 120001, 120011
Offset: 1

Views

Author

N. J. A. Sloane, Jan 13 2001

Keywords

Comments

Church's table extends through degree 7.

Examples

			The first few are x+1; x^2+x+2, x^2+2x+2; ...

Links

Crossrefs

Cf. A058944.
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.

Programs

  • Mathematica
    car = 3; maxDegree = 8;
okQ[{1, 1}] = True;
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 09 2019 *)

Extensions

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

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jan 13 2001

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

N. J. A. Sloane, Jan 13 2001

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Extensions

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

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

Original entry on oeis.org

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

Views

Author

Vladeta Jovovic, Feb 06 2001

Keywords

Comments

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

Links

Crossrefs

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.

Programs

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

Formula

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).
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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