A001693 -id:A001693 - cp's OEIS frontend

A058822 a(0) = 1, a(1) = 7; for n>=2 a(n) is the number of degree-n monic reducible polynomials over GF(7), i.e., a(n) = 7^n - A001693(n).

1, 7, 28, 231, 1813, 13447, 98105, 705895, 5044501, 35869911, 254229409, 1797569767, 12687856601, 89436009607, 629778626473, 4431057410423, 31155872769301, 218946366105607, 1537946178052697, 10798953333511399, 75802652996855281, 531948441984119239, 3732101910100912537

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 7-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A000420, A001693, A058766.

a[n_] := 7^n - DivisorSum[n, MoebiusMu[n/#] * 7^# &] / n; a[0] = 1; a[1] = 7; Array[a, 23, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 7^n, 7^n - sumdiv(n, d, moebius(d)*7^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

More terms from Michel Marcus, Oct 30 2017

A074650 Table T(n,k) read by downward antidiagonals: number of Lyndon words (aperiodic necklaces) with n beads of k colors, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 8, 3, 0, 6, 10, 20, 18, 6, 0, 7, 15, 40, 60, 48, 9, 0, 8, 21, 70, 150, 204, 116, 18, 0, 9, 28, 112, 315, 624, 670, 312, 30, 0, 10, 36, 168, 588, 1554, 2580, 2340, 810, 56, 0, 11, 45, 240, 1008, 3360, 7735, 11160, 8160, 2184, 99, 0

Offset: 1

Christian G. Bower, Aug 28 2002

D. E. Knuth uses the term 'prime strings' for Lyndon words because of the fundamental theorem stating the unique factorization of strings into nonincreasing prime strings (see Knuth 7.2.1.1). With this terminology T(n,k) is the number of k-ary n-tuples (a_1,...,a_n) such that the string a_1...a_n is prime. - Peter Luschny, Aug 14 2012
Also, for k a power of a prime, the number of monic irreducible polynomials of degree n over GF(k). - Andrew Howroyd, Dec 23 2017
An equivalent description: Array read by antidiagonals: T(n,k) = number of conjugacy classes of primitive words of length k >= 1 over an alphabet of size n >= 1.
There are a few incorrect values in Table 1 in the Perrin-Reutenauer paper (Christophe Reutenauer, personal communication), see A294438. - Lars Blomberg, Dec 05 2017
The fact that T(3,4) = 20 coincides with the number of the amino acids encoded by DNA made Francis Crick, John Griffith and Leslie Orgel conjecture in 1957 that the genetic code is a comma-free code, which later turned out to be false. [Hayes] - Andrey Zabolotskiy, Mar 24 2018

			T(4, 3) counts the 18 ternary prime strings of length 4 which are: 0001, 0002, 0011, 0012, 0021, 0022, 0102, 0111, 0112, 0121, 0122, 0211, 0212, 0221, 0222, 1112, 1122, 1222.
Square array starts:
  1,  2,   3,    4,     5,     6,      7, ...
  0,  1,   3,    6,    10,    15,     21, ...
  0,  2,   8,   20,    40,    70,    112, ...
  0,  3,  18,   60,   150,   315,    588, ...
  0,  6,  48,  204,   624,  1554,   3360, ...
  0,  9, 116,  670,  2580,  7735,  19544, ...
  0, 18, 312, 2340, 11160, 39990, 117648, ...
  ...
The transposed array starts:
   1  0  0     0     0      0       0        0         0          0,
   2  1  2     3     6      9      18       30        56         99,
   3  3  8    18    48    116     312      810      2184       5880,
   4  6  20   60   204    670    2340     8160     29120     104754,
   5 10  40  150   624   2580   11160    48750    217000     976248,
   6 15  70  315  1554   7735   39990   209790   1119720    6045837,
   7 21 112  588  3360  19544  117648   720300   4483696   28245840,
   8 28 168 1008  6552  43596  299592  2096640  14913024  107370900,
   9 36 240 1620 11808  88440  683280  5380020  43046640  348672528,
  10 45 330 2475 19998 166485 1428570 12498750 111111000  999989991,
  11 55 440 3630 32208 295020 2783880 26793030 261994040 2593726344,
  12 66 572 5148 49764 497354 5118828 53745120 573308736 6191711526,
  ...
The initial antidiagonals are:
   1
   2  0
   3  1   0
   4  3   2    0
   5  6   8    3    0
   6 10  20   18    6     0
   7 15  40   60   48     9     0
   8 21  70  150  204   116    18     0
   9 28 112  315  624   670   312    30     0
  10 36 168  588 1554  2580  2340   810    56    0
  11 45 240 1008 3360  7735 11160  8160  2184   99   0
  12 55 330 1620 6552 19544 39990 48750 29120 5880 186 0

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 97 (2.3.74)
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 495.
D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, pp. 26-27, Addison-Wesley, 2005.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
B. Hayes, The invention of the genetic code, American Scientist, Vol. 86, No. 1 (January-February 1998), pp. 8-14.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Irem Kucukoglu and Yilmaz Simsek, On k-ary Lyndon words and their generating functions, AIP Conference Proceedings 1863, 300004 (2017).
R. C. Lyndon, On Burnside's problem, Transactions of the American Mathematical Society 77, (1954) 202-215.
Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, arXiv preprint arXiv:1609.05438 [math.CO] (2016).
Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, Discrete Math., 341 (2018), 232-243.
Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, Discrete Math., 341 (2018), 232-243. [Annotated scanned copy of page 236 only.]
Wikipedia, Lyndon word
Index entries for sequences related to Lyndon words

Columns k: A001037 (k=2), A027376 (k=3), A027377 (k=4), A001692 (k=5), A032164 (k=6), A001693 (k=7), A027380 (k=8), A027381 (k=9), A032165 (k=10), A032166 (k=11), A032167 (k=12), A060216 (k=13), A060217 (k=14), A060218 (k=15), A060219 (k=16), A060220 (k=17), A060221 (k=18), A060222 (k=19).
Rows n: A000027 (n=1), A000217(k-1) (n=2), A007290(k+1) (n=3), A006011 (n=4), A208536(k+1) (n=5), A292350 (n=6), A208537(k+1) (n=7).
Cf. A000010, A008683, A075147 (main diagonal), A102659, A215474 (preprime strings), A383011.

t:= func< n,k | (&+[MoebiusMu(Floor(n/d))*k^d: d in Divisors(n)])/n >; // array
A074650:= func< n,k | t(k, n-k+1) >; // downward diagonals
[A074650(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Aug 01 2024

with(numtheory):
T:= proc(n, k) add(mobius(n/d)*k^d, d=divisors(n))/n end:
seq(seq(T(i, 1+d-i), i=1..d), d=1..11);  # Alois P. Heinz, Mar 28 2008

max = 12; t[n_, k_] := Total[ MoebiusMu[n/#]*k^# & /@ Divisors[n]]/n; Flatten[ Table[ t[n-k+1, k], {n, 1, max}, {k, n, 1, -1}]] (* Jean-François Alcover, Oct 18 2011, after Maple *)

T(n,k)=sumdiv(n,d,moebius(n/d)*k^d)/n \\ Charles R Greathouse IV, Oct 18 2011

# This algorithm generates and counts all k-ary n-tuples (a_1,..,a_n) such
# that the string a_1...a_n is prime. It is algorithm F in Knuth 7.2.1.1.
def A074650(n, k):
    a = [0]*(n+1); a[0]=-1
    j = 1; count = 0
    while(j != 0) :
        if j == n : count += 1; # print("".join(map(str,a[1:])))
        else: j = n
        while a[j] >= k-1 : j -= 1
        a[j] += 1
        for i in (j+1..n): a[i] = a[i-j]
    return count   # Peter Luschny, Aug 14 2012

T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*k^d.
T(n,k) = (k^n - Sum_{dAlois P. Heinz, Mar 28 2008
From Richard L. Ollerton, May 10 2021: (Start)
T(n,k) = (1/n)*Sum_{i=1..n} mu(gcd(n,i))*k^(n/gcd(n,i))/phi(n/gcd(n,i)).
T(n,k) = (1/n)*Sum_{i=1..n} mu(n/gcd(n,i))*k^gcd(n,i)/phi(n/gcd(n,i)). (End)
From Seiichi Manyama, Apr 12 2025: (Start)
G.f. of column k: -Sum_{j>=1} mu(j) * log(1 - k*x^j) / j.
Product_{n>=1} 1/(1 - x^n)^T(n,k) = 1/(1 - k*x). (End)

A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

Original entry on oeis.org

1, 3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880, 16104, 44220, 122640, 341484, 956576, 2690010, 7596480, 21522228, 61171656, 174336264, 498111952, 1426403748, 4093181688, 11767874940, 33891544368, 97764009000, 282429535752, 817028131140, 2366564736720, 6863037256208, 19924948267224, 57906879556410

Offset: 0

N. J. A. Sloane

Number of Lyndon words of length n on {1,2,3}. A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts. - John W. Layman, Jan 24 2006
Exponents in an expansion of the Hardy-Littlewood constant Product(1 - (3*p - 1)/(p - 1)^3, p prime >= 5), whose decimal expansion is in A065418: the constant equals Product_{n >= 2} (zeta(n)*(1 - 2^(-n))*(1 - 3^(-n)))^(-a(n)). - Michael Somos, Apr 05 2003
Number of aperiodic necklaces with n beads of 3 colors. - Herbert Kociemba, Nov 25 2016
Number of irreducible harmonic polylogarithms, see page 299 of Gehrmann and Remiddi reference and table 1 of Maître article. - F. Chapoton, Aug 09 2021
For n>=2, a(n) is the number of Hesse loops of length 2*n, see Theorem 22 of Dutta, Halbeisen, Hungerbühler link. - Sayan Dutta, Sep 22 2023
For n>=2, a(n) is the number of orbits of size n of isomorphism classes of elliptic curves under the Hesse derivative, see Theorem 2 of Kettinger link. - Jake Kettinger, Aug 07 2024

			For n = 2 the a(2)=3 polynomials are  x^2+1, x^2+x+2, x^2+2*x+2. - _Robert Israel_, Dec 16 2015

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..2102 (terms 0..200 from T. D. Noe)
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 4th line of Table 1 (p. 6).
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 6.
Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023.
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms. Comput. Phys. Comm. 141 (2001), no. 2, 296-312.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. See Table A.2.
Jake Kettinger, The dynamics of the Hesse derivative on the j-invariant, arXiv:2408.04117 [math.AG], 2024.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Computer Physics Communications, Volume 174, Issue 3, 1 February 2006, Pages 222-240.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag, 1978.
Index entries for sequences related to Lyndon words

Column 3 of A074650.

Cf. A000031, A001037, A001693, A001867, A027375, A027377, A054718, A102660.

with(numtheory): A027376 := n -> `if`(n = 0, 1,
add(mobius(d)*3^(n/d), d = divisors(n))/n):
seq(A027376(n), n = 0..32);

a[0]=1; a[n_] := Module[{ds=Divisors[n], i}, Sum[MoebiusMu[ds[[i]]]3^(n/ds[[i]]), {i, 1, Length[ds]}]/n]
a[0]=1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*3^#&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 01 2015 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,3],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n<1,n==0,sumdiv(n,d,moebius(n/d)*3^d)/n)

a(n) = (1/n)*Sum_{d|n} mu(d)*3^(n/d).
(1 - 3*x) = Product_{n>0} (1 - x^n)^a(n).
G.f.: k = 3, 1 - Sum_{i >= 1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) ~ 3^n / n. - Vaclav Kotesovec, Jul 01 2018
a(n) = 2*A046211(n) + A046209(n). - R. J. Mathar, Oct 21 2021

A027380 Number of irreducible polynomials of degree n over GF(8); dimensions of free Lie algebras.

Original entry on oeis.org

1, 8, 28, 168, 1008, 6552, 43596, 299592, 2096640, 14913024, 107370900, 780903144, 5726600880, 42288908760, 314146029564, 2345624803704, 17592184995840, 132458812569720, 1000799909722368, 7585009898729256

Offset: 0

N. J. A. Sloane

nonn

Number of Lyndon words with 8 letters. - Joerg Arndt, Jul 29 2014

Number of aperiodic necklaces with n beads of 8 colors. - Herbert Kociemba, Nov 25 2016

			G.f. = 1 + 8*x + 28*x^2 + 168*x^3 + 1008*x^4 + 6552*x^5 + 43596*x^6 + ...

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 0..200 from T. D. Noe)
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
Index entries for sequences related to Lyndon words

Column 8 of A074650.

Cf. A001037, A001693.

A027380 := proc(n)
    local d;
    if n = 0 then
        1;
    else
        add( 8^(n/d)*numtheory[mobius](d),d=numtheory[divisors](n)) ;
        %/n ;
    end if;
end proc: # R. J. Mathar, Jun 09 2016

f[n_] := (1/n)*Sum[MoebiusMu[d]*8^(n/d), {d, Divisors[n]}]; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Jul 28 2014 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,8],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n) = if(n, sumdiv(n, d, moebius(d)*8^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015

G.f.: k=8, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

a(n) = Sum_{d|n} mu(d)*8^(n/d)/n for n > 0. - Andrew Howroyd, Oct 13 2017

cp's OEIS Frontend

A058822 a(0) = 1, a(1) = 7; for n>=2 a(n) is the number of degree-n monic reducible polynomials over GF(7), i.e., a(n) = 7^n - A001693(n).

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A074650 Table T(n,k) read by downward antidiagonals: number of Lyndon words (aperiodic necklaces) with n beads of k colors, n >= 1, k >= 1.

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A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

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A027380 Number of irreducible polynomials of degree n over GF(8); dimensions of free Lie algebras.

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