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 1-4 of 4 results.

A054662 Number of monic irreducible polynomials over GF(5) with fixed nonzero trace.

Original entry on oeis.org

1, 2, 8, 30, 125, 516, 2232, 9750, 43400, 195250, 887784, 4068740, 18780048, 87191964, 406901000, 1907343750, 8975758272, 42385503300, 200773540296, 953674218750, 4541306267856, 21674415838068, 103660251783288
Offset: 1

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Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of 5-ary Lyndon words with trace 1 mod 5; trace 2 mod 5; trace 3 mod 5; trace 4 mod 5; also number of 5-ary Lyndon words of trace 1 over GF(5), trace 2 over GF(5); trace 3 over GF(5); trace 4 over GF(5).

Links

Crossrefs

Column 5 of A110540.
Cf. A000048, A051841, A046211, A046209, A054661, etc.

Programs

  • PARI
    a(n) = sumdiv(n, d, (gcd(d, 5)==1)*(moebius(d)*5^(n/d)))/(5*n); \\ Seiichi Manyama, May 29 2024

Formula

a(n) = 1/(5*n) * Sum_{d|n, gcd(d,5)=1} mu(d) * 5^(n/d). - Seiichi Manyama, May 29 2024

Extensions

More terms from James Sellers, Apr 19 2000

A300674 Number of monic irreducible polynomials of degree n over GF(8) that have a given nonzero trace.

Original entry on oeis.org

1, 4, 21, 128, 819, 5460, 37449, 262144, 1864128, 13421772, 97612893, 715827840, 5286113595, 39268272420, 293203100463, 2199023255552, 16557351571215, 125099989647360, 948126237341157, 7205759403792768, 54901024028884989, 419244183493398900, 3208129404123400281, 24595658764945981440
Offset: 1

Views

Author

Seiichi Manyama, Mar 11 2018

Keywords

Links

Crossrefs

Column 8 of A110540.
Cf. A000048, A008683, A054660, A300675.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[#] * 8^(n/#) &, OddQ[#] &] / (8*n); Array[a, 24] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    a(n) = sumdiv(n, d, if (d%2, moebius(d)*8^(n/d)))/(8*n); \\ Michel Marcus, Mar 11 2018

Formula

a(n) = (1/(8*n)) * Sum_{odd d divides n} mu(d)*8^(n/d), where mu is the Möbius function A008683.

A300675 Number of monic irreducible polynomials of degree n over GF(16) that have a given nonzero trace.

Original entry on oeis.org

1, 8, 85, 1024, 13107, 174760, 2396745, 33554432, 477218560, 6871947672, 99955602525, 1466015503360, 21651921285435, 321685687669320, 4803839602524143, 72057594037927936, 1085102592571150095, 16397105843297320960, 248545604361560274405, 3777893186295716170752, 57567896172125197996605
Offset: 1

Views

Author

Seiichi Manyama, Mar 11 2018

Keywords

Links

Crossrefs

Column 16 of A110540.
Cf. A000048, A008683, A054660, A300674.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[#] * 16^(n/#) &, OddQ[#] &] / (16*n); Array[a, 21] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    a(n) = sumdiv(n, d, if (d%2, moebius(d)*16^(n/d)))/(16*n); \\ Michel Marcus, Mar 11 2018

Formula

a(n) = (1/(16*n)) * Sum_{odd d divides n} mu(d)*16^(n/d), where mu is the Möbius function A008683.

A373277 Number of monic irreducible polynomials over GF(7) with fixed nonzero trace.

Original entry on oeis.org

1, 3, 16, 84, 480, 2792, 16807, 102900, 640528, 4035120, 25679568, 164775800, 1064714400, 6920635197, 45214871360, 296722542900, 1954878268800, 12923917122536, 85705978837392, 569944757250960, 3799631728452112, 25388448354582216, 169992219503608176
Offset: 1

Views

Author

Seiichi Manyama, May 29 2024

Keywords

Links

Crossrefs

Column 7 of A110540.

Programs

  • PARI
    a(n) = sumdiv(n, d, (gcd(d, 7)==1)*(moebius(d)*7^(n/d)))/(7*n);

Formula

a(n) = 1/(7*n) * Sum_{d|n, gcd(d,7)=1} mu(d) * 7^(n/d).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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