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 1-7 of 7 results.

A318622 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(2) (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 2, 8, 2, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 4, 1, 2, 6, 16, 2, 2, 2, 2, 1, 1, 2, 4, 2, 2, 3, 2, 2, 2, 2, 8, 2, 1, 4, 2, 1, 1, 2, 8, 2, 1, 1, 4, 1, 6, 6, 32, 4, 2, 1, 4, 2, 2, 2, 4, 8, 1, 2, 2, 2, 2, 2, 8, 1, 2, 1, 4, 8, 3, 2, 4, 8, 2, 6, 4, 6, 2, 2, 16, 2, 2
Offset: 1

Views

Author

Robert Israel, Aug 30 2018

Keywords

Comments

From Jianing Song, Sep 13 2022: (Start)
a(n) is also the number of irreducible factors in the factorization of the ideal (2) in Z[zeta_n], zeta_n = exp(2*Pi*i/n). Actually, if the n-th cyclotomic polynomial factors as Product_{i=1..a(n)} F_i(x) over GF(2), then the factorization of (2) in Z[zeta_m] is (p) = Product_{i=1..T(n,m)} (2,F_i(zeta_m)). See Page 47-48, Proposition 8.3 and Page 61-62, Proposition 10.3 of the Neukirch link for a proof; see also A327818.
As a result, 2 remains inert in Q(zeta_n) <=> a(n) = 1, which happens if and only if either n is odd, 2 is a primitive root modulo n, or n == 2 (mod 4), 2 is a primitive root modulo n/2.
Example 1: Phi_8(x) = x^4+1 == (x+1)^4 (mod 2), so in Z[zeta_8] = Z[i,sqrt(2)] we have (2) = (2,(zeta_8)+1)^4 = ((zeta_8)+1)^4. In fact we have 2 = -i*(3-2*sqrt(2)) * ((zeta_8)+1)^4.
Example 2: Phi_12(x) = x^4-x^2+1 == (x^2+x+1)^2 (mod 2), so in Z[zeta_12] = Z[i,sqrt(3)] we have (2) = (2,(zeta_12)^2+(zeta_12)+1)^2 = ((zeta_12)^2+(zeta_12)+1)^2. In fact we have 2 = (2-sqrt(3)) * (1-sqrt(-3))/2 * ((zeta_12)^2+(zeta_12)+1)^2. (End)

Links

Crossrefs

Cf. A000010, A000265, A002326, A091248, A129832 (a(n)=1).
Row 1 of A327818.

Programs

  • Maple
    f:= proc(n) option remember;  numtheory:-phi(n)/numtheory:-order(2, n/2^padic:-ordp(n, 2)) end proc:
map(f, [$1..200]);
  • Mathematica
    a[n_] := EulerPhi[n]/MultiplicativeOrder[2, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Jean-François Alcover, Apr 27 2019 *)
  • PARI
    a(n) = eulerphi(n)/znorder(Mod(2, (n >> valuation(n, 2)))); \\ Michel Marcus, Apr 27 2019

Formula

a(n) = A000010(n)/A002326((A000265(n)-1)/2).
A091248(n) = Sum_{d|n} a(d).

A327813 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(4) (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 8, 4, 2, 2, 4, 4, 2, 2, 8, 2, 2, 2, 4, 2, 4, 6, 16, 4, 4, 4, 4, 2, 2, 4, 8, 4, 4, 6, 4, 4, 2, 2, 16, 2, 2, 8, 4, 2, 2, 4, 8, 4, 2, 2, 8, 2, 6, 12, 32, 8, 4, 2, 8, 4, 4, 2, 8, 8, 2, 4, 4, 4, 4, 2, 16, 2, 4, 2, 8, 16, 6, 4, 8, 8, 4
Offset: 1

Views

Author

Jianing Song, Sep 26 2019

Keywords

Examples

			Let GF(4) = GF(2)[w], where w^2 + w + 1 = 0. Factorizations of the n-th cyclotomic polynomial over GF(4) for n <= 10:
n = 1: x + 1;
n = 2: x + 1;
n = 3: (x + w)*(x + (w+1));
n = 4: (x + 1)^2;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: (x + w)*(x + (w+1));
n = 7: (x^3 + x + 1)*(x^3 + x^2 + 1);
n = 8: (x + 1)^4;
n = 9: (x^3 + w)*(x^3 + (w+1));
n = 10: x^4 + x^3 + x^2 + x + 1.

Links

Crossrefs

Cf. A000010.
Row 3 of A327818.

Programs

  • Mathematica
    a[n_] := EulerPhi[n] / MultiplicativeOrder[4, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
  • PARI
    a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(4, s))

Formula

Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(4,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

A327814 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(5) (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 1, 2, 1, 4, 2, 2, 3, 1, 4, 2, 1, 1, 2, 8, 2, 2, 1, 4, 20, 3, 1, 2, 2, 4, 10, 2, 2, 1, 4, 2, 1, 2, 6, 8, 2, 2, 1, 4, 4, 1, 1, 4, 1, 20, 2, 6, 1, 1, 8, 4, 2, 2, 2, 8, 2, 10, 6, 2, 12, 2, 3, 2, 2, 4, 14, 4, 1, 1, 20, 4, 2, 6, 2, 8, 1, 2, 1, 4, 4, 1, 4, 4, 2, 4
Offset: 1

Views

Author

Jianing Song, Sep 26 2019

Keywords

Examples

			Factorizations of the n-th cyclotomic polynomial over GF(5) for n <= 10:
n = 1: x - 1;
n = 2: x + 1;
n = 3: x^2 + x + 1;
n = 4: (x + 2)*(x - 2);
n = 5: (x - 1)^4;
n = 6: x^2 - x + 1;
n = 7: x^6 + x^5 + x^4 + x^3 + x^2 + x + 1;
n = 8: (x^2 + 2)*(x^2 - 2);
n = 9: x^6 + x^3 + 1;
n = 10: (x + 1)^4.

Links

Crossrefs

Cf. A000010.
Row 4 of A327818.

Programs

  • Mathematica
    a[n_] := EulerPhi[n] / MultiplicativeOrder[5, n / 5^IntegerExponent[n, 5]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
  • PARI
    a(n) = my(s=n/5^valuation(n, 5)); eulerphi(n)/znorder(Mod(5, s))

Formula

Let n = 5^e*s, gcd(5,s) = 1, then a(n) = phi(n)/ord(5,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

A327815 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(7) (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 6, 2, 2, 1, 1, 2, 1, 6, 2, 4, 1, 2, 6, 2, 12, 1, 1, 4, 5, 1, 2, 6, 4, 2, 2, 4, 2, 1, 6, 2, 4, 6, 2, 4, 1, 12, 7, 2, 2, 1, 2, 8, 42, 5, 2, 2, 2, 2, 2, 12, 12, 4, 2, 4, 1, 2, 12, 4, 4, 2, 1, 2, 2, 6, 1, 4, 3, 4, 10, 6, 6, 2, 1, 8, 2, 1, 2, 12, 4, 7, 8, 4, 1, 2
Offset: 1

Views

Author

Jianing Song, Sep 26 2019

Keywords

Examples

			Factorizations of the n-th cyclotomic polynomial over GF(7) for n <= 10:
n = 1: x - 1;
n = 2: x + 1;
n = 3: (x + 3)*(x - 2);
n = 4: x^2 + 1;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: (x + 2)*(x - 3);
n = 7: (x - 1)^6;
n = 8: (x^2 + 3x + 1)*(x^2 - 3x + 1);
n = 9: (x^3 + 3)*(x^3 - 2);
n = 10: x^4 - x^3 + x^2 - x^2 + 1.

Links

Crossrefs

Cf. A000010.
Row 5 of A327818.

Programs

  • Mathematica
    a[n_] := EulerPhi[n] / MultiplicativeOrder[7, n / 7^IntegerExponent[n, 7]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
  • PARI
    a(n) = my(s=n/7^valuation(n, 7)); eulerphi(n)/znorder(Mod(7, s))

Formula

Let n = 7^e*s, gcd(7,s) = 1, then a(n) = phi(n)/ord(7,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

A327816 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(8) (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 4, 3, 1, 1, 2, 3, 6, 2, 8, 2, 3, 3, 2, 6, 1, 2, 4, 1, 3, 3, 12, 1, 2, 6, 16, 2, 2, 6, 6, 3, 3, 6, 4, 2, 6, 3, 2, 6, 2, 2, 8, 6, 1, 4, 6, 1, 3, 2, 24, 6, 1, 1, 4, 3, 6, 18, 32, 12, 2, 3, 4, 2, 6, 2, 12, 24, 3, 2, 6, 6, 6, 6, 8, 3, 2, 1, 12, 8, 3, 2, 4, 8, 6
Offset: 1

Views

Author

Jianing Song, Sep 26 2019

Keywords

Examples

			Let GF(8) = GF(2)[y]/(y^3+y+1). Factorizations of the n-th cyclotomic polynomial over GF(8) for n <= 10:
n = 1: x + 1;
n = 2: x + 1;
n = 3: x^2 + x + 1;
n = 4: (x + 1)^2;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: x^2 + x + 1;
n = 7: (x + y)*(x + (y+1))*(x + y^2)*(x + (y^2+1))*(x + (y^2+y))*(x + (y^2+y+1));
n = 8: (x + 1)^4;
n = 9: (x^2 + y*x + 1)*(x^2 + (y+1)*x + 1)*(x^2 + y^2*x + 1);
n = 10: x^4 + x^3 + x^2 + x + 1.

Links

Crossrefs

Cf. A000010.
Row 6 of A327818.

Programs

  • Mathematica
    a[n_] := EulerPhi[n] / MultiplicativeOrder[8, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
  • PARI
    a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(8, s))

Formula

Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(8,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

A327817 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(9) (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 4, 6, 2, 2, 4, 4, 2, 4, 4, 2, 6, 2, 4, 4, 2, 2, 8, 2, 4, 18, 4, 2, 4, 2, 4, 4, 2, 4, 12, 4, 2, 8, 8, 10, 4, 2, 4, 12, 2, 2, 8, 2, 2, 4, 8, 2, 18, 4, 8, 4, 2, 2, 8, 12, 2, 12, 4, 8, 4, 6, 4, 4, 4, 2, 24, 12, 4, 4, 4, 4, 8, 2, 16, 54, 10, 2, 8, 8, 2, 4, 8, 2, 12
Offset: 1

Views

Author

Jianing Song, Sep 26 2019

Keywords

Examples

			Let GF(9) = GF(3)[i], where i^2 = -1. Factorizations of the n-th cyclotomic polynomial over GF(9) for n <= 10:
n = 1: x - 1;
n = 2: x + 1;
n = 3: (x - 1)^2;
n = 4: (x + i)*(x - i);
n = 5: (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
n = 6: (x + 1)^2;
n = 7: (x^3 + (-1+i)*x^2 + (1+i)*x - 1)*(x^3 + (-1-i)*x^2 + (1-i)*x - 1);
n = 8: (x + (1+i))*(x + (1-i))*(x + (-1+i))*(x + (-1-i));
n = 9: (x - 1)^6;
n = 10: (x^2 + (1+i)*x + 1)*(x^2 + (1-i)*x + 1).

Links

Crossrefs

Cf. A000010.
Row 7 of A327818.

Programs

  • Mathematica
    a[n_] := EulerPhi[n] / MultiplicativeOrder[9, n / 3^IntegerExponent[n, 3]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
  • PARI
    a(n) = my(s=n/3^valuation(n, 3)); eulerphi(n)/znorder(Mod(9, s))

Formula

Let n = 3^e*s, gcd(3,s) = 1, then a(n) = phi(n)/ord(9,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

A327812 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(3) (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 6, 1, 2, 2, 4, 1, 2, 2, 1, 6, 1, 2, 2, 2, 2, 4, 1, 4, 18, 2, 1, 2, 1, 2, 4, 1, 2, 6, 2, 1, 8, 4, 5, 2, 1, 2, 6, 2, 2, 4, 1, 1, 2, 4, 1, 18, 2, 4, 2, 1, 2, 4, 6, 1, 6, 2, 4, 4, 3, 2, 4, 2, 2, 12, 6, 2, 2, 2, 2, 8, 1, 8, 54, 5, 2, 4, 4, 1, 2, 4, 1, 6
Offset: 1

Views

Author

Jianing Song, Sep 26 2019

Keywords

Examples

			Factorizations of the n-th cyclotomic polynomial over GF(3) for n <= 10:
n = 1: x - 1;
n = 2: x + 1;
n = 3: (x - 1)^2;
n = 4: x^2 + 1;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: (x + 1)^2;
n = 7: x^6 + x^5 + x^4 + x^3 + x^2 + x + 1;
n = 8: (x^2 + x - 1)*(x^2 - x - 1);
n = 9: (x - 1)^6;
n = 10: x^4 - x^3 + x^2 - x^2 + 1.

Crossrefs

Cf. A000010.
Row 2 of A327818.

Programs

  • PARI
    a(n) = my(s=n/3^valuation(n, 3)); eulerphi(n)/znorder(Mod(3, s))

Formula

Let n = 3^e*s, gcd(3,s) = 1, then a(n) = phi(n)/ord(3,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
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