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.

A152049 Number of conjugacy classes of primitive elements in GF(2^n) which have trace 0.

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 9, 9, 23, 29, 89, 72, 315, 375, 899, 1031, 3855, 3886, 13797, 12000, 42328, 59989, 178529, 138256, 647969, 859841, 2101143, 2370917, 9204061, 8911060, 34636833, 33556537, 105508927, 168423669, 464635937
Offset: 1

Views

Author

David A. Madore, Nov 21 2008

Keywords

Comments

Also number of primitive polynomials of degree n over GF(2) whose second-highest coefficient is 0.
Always less than A011260 (and exactly one half of it when 2^n-1 is prime).

Examples

			a(3)=1 because of the two primitive degree 3 polynomials over GF(2), namely t^3+t+1 and t^3+t^2+1, only the former has a zero next-to-highest coefficient.
Similarly, a(13)=315, because of half (4096) of the 8192 elements of GF(2^13) have trace 0 and all except 0 (since 1 has trace 1) are primitive, so there are 4095/13=315 conjugacy classes of primitive elements of trace 0.

Crossrefs

Cf. A192507 (GF(3^n)), A192508 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

Programs

  • GAP
    a := function(n)
    local q,k,cnt,x; q:=2^n; k:=GF(2,n); cnt:=0;
    for x in k do
        if Trace(k, GF(2), x)=0*Z(2) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..32] do  Print (a(n), ", ");  od;

Formula

a(n) = A192211(n)/n. [Joerg Arndt, Jul 03 2011]

Extensions

More terms (13797...8911060) by Joerg Arndt, Jun 26 2011.
More terms (34636833...464635937) by Joerg Arndt, Jul 03 2011.

A192212 Number of zero trace primitive elements in Galois field GF(3^n).

Original entry on oeis.org

0, 0, 3, 8, 35, 84, 364, 832, 2997, 8700, 28281, 55080, 265720, 794304, 2006730, 5240928, 21511647, 47028492, 193587884, 415388400, 1607804226
Offset: 1

Views

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

Crossrefs

Cf. A192211, A192213, A192214, A192215, A192216 for other primes

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(3^n):
        if x!=0 and x.trace()==0 and x.multiplicative_order()==3^n-1: ans += 1
    return ans  # Robin Visser, Apr 26 2024

Formula

a(n) = n * A192507(n). [Joerg Arndt, Oct 03 2012]

Extensions

a(12)-a(17) from Joerg Arndt, Oct 03 2012
a(18)-a(21) from Robin Visser, Apr 26 2024

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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