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

A192213 Number of zero trace primitive elements in Galois field GF(5^n).

Original entry on oeis.org

0, 0, 12, 32, 270, 840, 7812, 23808, 178452, 583880, 4882812, 12434160, 122070312, 391954136, 2630952180
Offset: 1

Views

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

Crossrefs

Cf. A192508.
Cf. A192211, A192212, A192214, A192215, A192216 for other primes.

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(5^n):
        if x!=0 and x.trace()==0 and x.multiplicative_order()==5^n-1: ans += 1
    return ans  # Robin Visser, May 10 2024

Formula

a(n) = n * A192508(n). - Joerg Arndt, Jul 03 2011

Extensions

a(9)-a(11) from Joerg Arndt, Oct 03 2012
a(12)-a(15) from Robin Visser, May 10 2024

A192214 Number of zero trace primitive elements in Galois field GF(7^n).

Original entry on oeis.org

0, 0, 9, 80, 800, 5076, 37982, 218880, 1770660, 12155480, 94076488, 447960240
Offset: 1

Views

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

Crossrefs

Cf. A192211, A192212, A192213, A192215, A192216 for other primes.
Cf. A192509.

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(7^n):
        if x!=0 and x.trace()==0 and x.multiplicative_order()==7^n-1: ans += 1
    return ans  # Robin Visser, Jun 01 2024

Formula

a(n) = n * A192509(n). - Joerg Arndt, Jul 03 2011

Extensions

a(7)-a(9) added using the data at A192509 by Amiram Eldar, May 03 2024
a(10)-a(12) from Robin Visser, Jun 01 2024

A192215 Number of zero trace primitive elements in Galois field GF(11^n).

Original entry on oeis.org

0, 0, 48, 320, 5925, 33936, 691880, 5107200, 69610716, 628484000
Offset: 1

Views

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

Crossrefs

Cf. A192211, A192212, A192213, A192214, A192216 for other primes.

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(11^n):
        if x!=0 and x.trace()==0 and x.multiplicative_order()==11^n-1: ans += 1
    return ans  # Robin Visser, May 10 2024

Formula

a(n) = n * A192510(n). - Joerg Arndt, Jul 03 2011

Extensions

a(6)-a(7) from A192510 by Jean-François Alcover, Mar 02 2020
a(8)-a(10) from Robin Visser, May 10 2024

A192216 Number of zero trace primitive elements in Galois field GF(13^n).

Original entry on oeis.org

0, 0, 54, 448, 9520, 103104, 1608936, 13488064, 267423822
Offset: 1

Views

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

Crossrefs

Cf. A192211, A192212, A192213, A192214, A192215 for other primes.
Cf. A192511.

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(13^n):
        if x!=0 and x.trace()==0 and x.multiplicative_order()==13^n-1: ans += 1
    return ans  # Robin Visser, Jun 01 2024

Formula

a(n) = n * A192511(n). - Joerg Arndt, Jul 03 2011

Extensions

a(6) added using the data at A192511 by Amiram Eldar, May 03 2024
a(7)-a(9) from Robin Visser, Jun 01 2024
Showing 1-6 of 6 results.

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