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.

User: Pasha Zusmanovich

Pasha Zusmanovich's wiki page.

Pasha Zusmanovich has authored 10 sequences.

A245490 a(n) = n^(n^3).

Original entry on oeis.org

1, 1, 256, 7625597484987, 340282366920938463463374607431768211456
Offset: 0

Author

Pasha Zusmanovich, Jul 23 2014

Keywords

Comments

Number of ternary operations on a set of n elements.
Next term has 88 decimal digits.

Programs

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

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

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

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

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

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

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

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

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.

Crossrefs

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

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

A161392 Dimension sequence of the alternative operad in characteristics different from 2 and 3.

Original entry on oeis.org

1, 2, 7, 32, 175, 1080
Offset: 1

Author

Pasha Zusmanovich, Jun 09 2009

Keywords

Crossrefs

A161391 Dimension sequence of the right-alternative (=left-alternative) operad.

Original entry on oeis.org

1, 2, 9, 60, 530, 5820
Offset: 1

Author

Pasha Zusmanovich, Jun 09 2009

Keywords

Crossrefs

A161393 Dimension sequence of the alternative operad in characteristic 3.

Original entry on oeis.org

1, 2, 7, 32, 175, 1081
Offset: 1

Author

Pasha Zusmanovich, Jun 09 2009

Keywords

Crossrefs