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 1-6 of 6 results.

A192507 Number of conjugacy classes of primitive elements in GF(3^n) which have trace 0.

Original entry on oeis.org

0, 0, 1, 2, 7, 14, 52, 104, 333, 870, 2571, 4590, 20440, 56736, 133782, 327558, 1265391, 2612694, 10188836, 20769420, 76562106
Offset: 1

Views

Author

Joerg Arndt, Jul 03 2011

Keywords

Comments

Also number of primitive polynomials of degree n over GF(3) whose second-highest coefficient is 0.

Crossrefs

Cf. A152049 (GF(2^n)), A192507 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).
Cf. A027385 (number of primitive polynomials of degree n over GF(3)).

Programs

  • GAP
    p := 3;
a := function(n)
    local q, k, cnt, x;
    q:=p^n;  k:=GF(p, n);  cnt:=0;
    for x in k do
        if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..16] do  Print (a(n), ", ");  od;
  • Sage
    # much more efficient
p=3; # choose characteristic
for n in range(1,66):
    F = GF(p^n, 'x')
    g = F.multiplicative_generator() # generator
    vt = vector(ZZ,p) # stats: trace
    m = p^n - 1 # size of multiplicative group
    # Compute all irreducible polynomials via Lyndon words:
    for w in LyndonWords(p,n): # digits of Lyndon words range form 1,..,p
        e = sum( (w[j]-1) * p^j for j in range(0,n) )
        if gcd(m, e) == 1: # primitive elements only
            f = g^e
            t = f.trace().lift(); # trace (over ZZ)
            vt[t] += 1
    print(vt[0]) # choose index 0,1,..,p-1 for different traces
# Joerg Arndt, Oct 03 2012

Formula

a(n) = A192212(n) / n.

Extensions

Added terms >=2571, Joerg Arndt, Oct 03 2012
a(18)-a(21) from Robin Visser, Apr 26 2024

A192211 Number of zero trace primitive elements in Galois field GF(2^n).

Original entry on oeis.org

0, 0, 3, 4, 15, 12, 63, 72, 207, 290, 979, 864, 4095, 5250, 13485, 16496, 65535, 69948, 262143, 240000, 888888, 1319758, 4106167, 3318144, 16199225, 22355866, 56730861, 66385676, 266917769, 267331800, 1073741823, 1073809184, 3481794591, 5726404746, 16262257795
Offset: 1

Views

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

References

  • R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997. Chapter 2 discusses primitivity in sections 1-2 and trace in section 3.

Links

Crossrefs

Cf. A192212, A192213, A192214, A192215, A192216 for other primes.
Cf. A152049, A107222.

Programs

  • GAP
    p := 2;
for n in [1..17] do
    F := GF(p^n);
    num := 0;
    for f in F do
        if (f = Zero(F)) then continue; fi;
        if (Trace(f) <> Zero(F)) then continue; fi;
        if (Order(f) <> Size(F) - 1) then continue; fi;
        num := num + 1;
    od;
    Print (num, ",");
od;

Formula

a(n) = n * A152049(n). [Joerg Arndt, Jul 03 2011]

Extensions

Terms 69948, ..., 1073809184 from Joerg Arndt, Jun 26 2011
Terms >1073809184 from Joerg Arndt, Jul 03 2011

A192509 Number of conjugacy classes of primitive elements in GF(7^n) which have trace 0.

Original entry on oeis.org

0, 0, 3, 20, 160, 846, 5426, 27360, 196740, 1215548, 8552408, 37330020
Offset: 1

Views

Author

Joerg Arndt, Jul 03 2011

Keywords

Comments

Also number of primitive polynomials of degree n over GF(7) whose second-highest coefficient is 0.

Crossrefs

Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192508 (GF(5^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

Programs

  • GAP
    p := 7;
a := function(n)
    local q, k, cnt, x;
    q:=p^n;  k:=GF(p, n);  cnt:=0;
    for x in k do
        if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..16] do  Print (a(n), ", ");  od;
  • Sage
    # See A192507 (change first line p=3 to p=7)

Formula

a(n) = A192214(n) / n.

Extensions

a(7)-a(9) from Joerg Arndt, Oct 14 2012
a(10)-a(12) from Robin Visser, Jun 01 2024

A192510 Number of conjugacy classes of primitive elements in GF(11^n) which have trace 0.

Original entry on oeis.org

0, 0, 16, 80, 1185, 5656, 98840, 638400, 7734524, 62848400
Offset: 1

Views

Author

Joerg Arndt, Jul 03 2011

Keywords

Comments

Also number of primitive polynomials of degree n over GF(11) whose second-highest coefficient is 0.

Crossrefs

Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192508 (GF(5^n)), A192509 (GF(7^n)), A192511 (GF(13^n)).

Programs

  • GAP
    p := 11;
a := function(n)
    local q, k, cnt, x;
    q:=p^n;  k:=GF(p, n);  cnt:=0;
    for x in k do
        if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..16] do  Print (a(n), ", ");  od;
  • Sage
    # See A192507 (change first line p=3 to p=11)

Formula

a(n) = A192215(n) / n.

Extensions

Added terms a(6) and a(7), Joerg Arndt, Oct 14 2012
a(8)-a(10) from Robin Visser, May 10 2024

A192511 Number of conjugacy classes of primitive elements in GF(13^n) which have trace 0.

Original entry on oeis.org

0, 0, 18, 112, 1904, 17184, 229848, 1686008, 29713758
Offset: 1

Views

Author

Joerg Arndt, Jul 03 2011

Keywords

Comments

Also number of primitive polynomials of degree n over GF(13) whose second-highest coefficient is 0.

Crossrefs

Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192508 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)).

Programs

  • GAP
    p := 13;
a := function(n)
    local q, k, cnt, x;
    q:=p^n;  k:=GF(p, n);  cnt:=0;
    for x in k do
        if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..16] do  Print (a(n), ", ");  od;
  • Sage
    # See A192507 (change first line p=3 to p=13)

Formula

a(n) = A192216(n) / n.

Extensions

a(7)-a(9) from Robin Visser, Jun 01 2024

A192508 Number of conjugacy classes of primitive elements in GF(5^n) which have trace 0.

Original entry on oeis.org

0, 0, 4, 8, 54, 140, 1116, 2976, 19828, 58388, 443892, 1036180, 9390024, 27996724, 175396812
Offset: 1

Views

Author

Joerg Arndt, Jul 03 2011

Keywords

Comments

Also number of primitive polynomials of degree n over GF(5) whose second-highest coefficient is 0.

Crossrefs

Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

Programs

  • GAP
    p := 5;
a := function(n)
    local q, k, cnt, x;
    q:=p^n;  k:=GF(p, n);  cnt:=0;
    for x in k do
        if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..16] do  Print (a(n), ", ");  od;
  • Sage
    # See A192507 (change first line p=3 to p=5)

Formula

a(n) = A192213(n) / n

Extensions

Added terms 19828..443892, Joerg Arndt, Oct 03 2012
a(12)-a(15) from Robin Visser, May 10 2024
Showing 1-6 of 6 results.

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