A102659 -id:A102659 - cp's OEIS frontend

A001037 Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.

1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008, 9586395, 18512790, 35790267, 69273666, 134215680, 260300986, 505286415, 981706806, 1908866960, 3714566310, 7233615333, 14096302710, 27487764474

Offset: 0

N. J. A. Sloane

Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence.
This sequence also represents the number N of cycles of length L in a digraph under x^2 seen modulo a Mersenne prime M_q=2^q-1. This number does not depend on q and L is any divisor of q-1. See Theorem 5 and Corollary 3 of the Shallit and Vasiga paper: N=sum(eulerphi(d)/order(d,2)) where d is a divisor of 2^(q-1)-1 such that order(d,2)=L. - Tony Reix, Nov 17 2005
Except for a(0) = 1, Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 7th (rightmost) column. Other columns of the table include (but are not identified as) A006206-A006208. - Jonathan Vos Post, Jun 18 2007
"Number of binary Lyndon words" means: number of binary strings inequivalent modulo rotation (cyclic permutation) of the digits and not having a period smaller than n. This provides a link to A103314, since these strings correspond to the inequivalent zero-sum subsets of U_m (m-th roots of unity) obtained by taking the union of U_n (n|m) with 0 or more U_d (n | d, d | m) multiplied by some power of exp(i 2Pi/n) to make them mutually disjoint. (But not all zero-sum subsets of U_m are of that form.) - M. F. Hasler, Jan 14 2007
Also the number of dynamical cycles of period n of a threshold Boolean automata network which is a quasi-minimal positive circuit of size a multiple of n and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Feb 25 2009
Also, the number of periodic points with (minimal) period n in the iteration of the tent map f(x):=2min{x,1-x} on the unit interval. - Pietro Majer, Sep 22 2009
Number of distinct cycles of minimal period n in a shift dynamical system associated with a totally disconnected hyperbolic iterated function system (see Barnsley link). - Michel Marcus, Oct 06 2013
From Jean-Christophe Hervé, Oct 26 2014: (Start)
For n > 0, a(n) is also the number of orbits of size n of the transform associated to the Kolakoski sequence A000002 (and this is true for any map with 2^n periodic points of period n). The Kolakoski transform changes a sequence of 1's and 2's by the sequence of the lengths of its runs. The Kolakoski sequence is one of the two fixed points of this transform, the other being the same sequence without the initial term. A025142 and A025143 are the periodic points of the orbit of size 2. A027375(n) = n*a(n) gives the number of periodic points of minimal period n.
For n > 1, this sequence is equal to A059966 and to A060477, and for n = 1, a(1) = A059966(1)+1 = A060477(1)-1; this because the n-th term of all 3 sequences is equal to (1/n)*sum_{d|n} mu(n/d)*(2^d+e), with e = -1/0/1 for resp. A059966/this sequence/A060477, and sum_{d|n} mu(n/d) equals 1 for n = 1 and 0 for all n > 1. (End)
Warning: A000031 and A001037 are easily confused, since they have similar formulas.
From Petros Hadjicostas, Jul 14 2020: (Start)
Following Kam Cheong Au (2020), let d(w,N) be the dimension of the Q-span of weight w and level N of colored multiple zeta values (CMZV). Here Q are the rational numbers.
Deligne's bound says that d(w,N) <= D(w,N), where 1 + Sum_{w >= 1} D(w,N)*t^w = (1 - a*t + b*t^2)^(-1) when N >= 3, where a = phi(N)/2 + omega(N) and b = omega(N) - 1 (with omega(N) = A001221(N) being the number of distinct primes of N).
For N = 3, a = phi(3)/2 + omega(3) = 2/2 + 1 = 2 and b = omega(3) - 1 = 0. It follows that D(w, N=3) = A000079(w) = 2^w.
For some reason, Kam Cheong Au (2020) assumes Deligne's bound is tight, i.e., d(w,N) = D(w,N). He sets Sum_{w >= 1} c(w,N)*t^w = log(1 + Sum_{w >= 1} d(w,N)*t^w) = log(1 + Sum_{w >= 1} D(w,N)*t^w) = -log(1 - a*t + b*t^2) for N >= 3.
For N = 3, we get that c(w, N=3) = A000079(w)/w = 2^w/w.
He defines d*(w,N) = Sum_{k | w} (mu(k)/k)*c(w/k,N) to be the "number of primitive constants of weight w and level N". (Using the terminology of A113788, we may perhaps call d*(w,N) the number of irreducible colored multiple zeta values at weight w and level N.)
Using standard techniques of the theory of g.f.'s, we can prove that Sum_{w >= 1} d*(w,N)*t^w = Sum_{s >= 1} (mu(s)/s) Sum_{k >= 1} c(k,N)*(t^s)^k = -Sum_{s >= 1} (mu(s)/s)*log(1 - a*t^s + b*t^(2*s)).
For N = 3, we saw that a = 2 and b = 0, and hence d*(w, N=3) = a(w) = Sum_{k | w} (mu(k)/k) * 2^(w/k) / (w/k) = (1/w) * Sum_{k | w} mu(k) * 2^(w/k) for w >= 1. See Table 1 on p. 6 in Kam Cheong Au (2020). (End)

			Binary strings (Lyndon words, cf. A102659):
a(0) = 1 = #{ "" },
a(1) = 2 = #{ "0", "1" },
a(2) = 1 = #{ "01" },
a(3) = 2 = #{ "001", "011" },
a(4) = 3 = #{ "0001", "0011", "0111" },
a(5) = 6 = #{ "00001", "00011", "00101", "00111", "01011", "01111" }.

Michael F. Barnsley, Fractals Everywhere, Academic Press, San Diego, 1988, page 171, Lemma 3.
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie. On the digraph defined by squaring mod m, when m has primitive roots. Congr. Numer. 82 (1991), 167-177.
P. J. Freyd and A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. See 1.925.
M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983, pp. 65, 79.
Robert M. May, "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019
Guy Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
M. R. Nester, (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in entries N0046 and N0287).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3333 (terms 0..200 from T. D. Noe)
Per Alexandersson and Joakim Uhlin, Cyclic sieving, skew Macdonald polynomials and Schur positivity, arXiv:1908.00083 [math.CO], 2019.
Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, and Mike Zabrocki, When are Hopf algebras determined by integer sequences?, arXiv:2505.06941 [math.CO], 2025. See p. 17.
Joerg Arndt, Matters Computational (The Fxtbook), pp.379-383, pp.843-845.
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 3rd line of Table 1 (p. 6).
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie, On a digraph defined by squaring modulo n, Fibonacci Quart. 30 (1992), 322-333.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Émilie Charlier, Manon Philibert, and Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. Also J. Comb. Thy. A, 167 (2019), 60-90.
R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.
J. Demongeot, M. Noual and S. Sene, On the number of attractors of positive and negative threshold Boolean automata circuits, 2010 IEEE 24th Intl. Conf. Advan. Inf. Network. Appl. Workshops (WAINA), p 782-789.
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.
Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, arXiv:0706.2297 [math.DS], 2007; Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.
S. V. Duzhin and D. V. Pasechnik, Groups acting on necklaces and sandpile groups, 2014. See page 85. - _N. J. A. Sloane_, Jun 30 2014
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
M. A. Harrison, On the classification of Boolean functions by the general linear and affine groups, J. Soc. Indust. Appl. Math. 12 (1964) 285-299.
A. Knopfmacher and M. E. Mays, Graph Compositions I: Basic enumeration, Integers 1 (2001), A4, eq. (1).
T. Laarhoven and B de Weger, The Collatz conjecture and De Bruijn graphs, arXiv preprint arXiv:1209.3495 [math.NT], 2012. - From _N. J. A. Sloane_, Dec 27 2012
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arithmetica, LVI (1990), pp. 33-53.
Piotr Laskawiec, Joint Spectral Radius with focus on pairs of 2 X 2 matrices, Ph. D. Dissertation, Penn. State Univ. (2025). See p. 55.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence gamma_{2,j}^(T), arXiv:0903.2514 [math.NT], 2009-2011.
Ueli M. Maurer, Asymptotically-tight bounds on the number of cycles in generalized de Bruijn-Good graphs, Discrete applied mathematics 37 (1992): 421-436. See Table 1.
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
H. Meyn and W. Götz, Self-reciprocal polynomials over finite fields, Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
J.-F. Michon and P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.
Hans Z. Munthe-Kaas and Jonatan Stava, Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces, arXiv:2306.15582 [math.DG], 2023.
Mathilde Noual, Dynamics of Circuits and Intersecting Circuits, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 2012, Volume 7183/2012, 433-444, ArXiv 1011.3930 [cs.DM]. - _N. J. A. Sloane_, Jul 07 2012
Cormac O'Sullivan, Topographs for binary quadratic forms and class numbers, arXiv:2408.14405 [math.NT], 2024. See p. 30.
George Petrides and Johannes Mykkeltveit, On the Classification of Periodic Binary Sequences into Nonlinear Complexity Classes, in Sequences and Their Applications SETA 2006, Lecture Notes in Computer Science, Volume 4086/2006, pp 209-222. [From _N. J. A. Sloane_, Jul 09 2009]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Primitive and Irreducible Polynomials
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Troy Vasiga and Jeffrey Shallit, On the iteration of certain quadratic maps over GF(p), Discrete Mathematics, Volume 277, Issues 1-3, 2004, pages 219-240.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
M. Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Lyndon Word
Wikipedia, Lyndon word
Index entries for sequences related to Lyndon words
Index entries for "core" sequences

Column 2 of A074650.
Row sums of A051168, which gives the number of Lyndon words with fixed number of zeros and ones.
Euler transform is A000079.
See A058943 and A102569 for initial terms. See also A058947, A011260, A059966.
Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.
Cf. A000031 (n-bead necklaces but may have period dividing n), A014580, A046211, A046209, A006206-A006208, A038063, A060477, A103314.
Cf. A027750, A008683, A254040.
See also A102659 for the list of binary Lyndon words themselves.
Cf. A000010, A008683.

a001037 0 = 1
a001037 n = (sum $ map (\d -> (a000079 d) * a008683 (n `div` d)) $
                       a027750_row n) `div` n
-- Reinhard Zumkeller, Feb 01 2013

with(numtheory): A001037 := proc(n) local a,d; if n = 0 then RETURN(1); else a := 0: for d in divisors(n) do a := a+mobius(n/d)*2^d; od: RETURN(a/n); fi; end;

f[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[n/d]*2^d/n)]; Array[f, 32]

A001037(n)=if(n>1,sumdiv(n,d,moebius(d)*2^(n/d))/n,n+1) \\ Edited by M. F. Hasler, Jan 11 2016

{a(n)=polcoeff(1-sum(k=1,n,moebius(k)/k*log(1-2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

a(n)=if(n>1,my(s);forstep(i=2^n+1,2^(n+1),2,s+=polisirreducible(Mod(1,2) * Pol(binary(i))));s,n+1) \\ Charles R Greathouse IV, Jan 26 2012

from sympy import divisors, mobius
def a(n): return sum(mobius(d) * 2**(n//d) for d in divisors(n))/n if n>1 else n + 1 # Indranil Ghosh, Apr 26 2017

For n >= 1:
a(n) = (1/n)*Sum_{d | n} mu(n/d)*2^d.
A000031(n) = Sum_{d | n} a(d).
2^n = Sum_{d | n} d*a(d).
a(n) = A027375(n)/n.
a(n) = A000048(n) + A051841(n).
For n > 1, a(n) = A059966(n) = A060477(n).
G.f.: 1 - Sum_{n >= 1} moebius(n)*log(1 - 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
From Richard L. Ollerton, May 10 2021: (Start)
For n >= 1:
a(n) = (1/n)*Sum_{k=1..n} mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
a(n) ~ 2^n / n. - Vaclav Kotesovec, Aug 11 2021

Revised by N. J. A. Sloane, Jun 10 2012

A275692 Numbers k such that every rotation of the binary digits of k is less than k.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 26, 28, 30, 32, 40, 48, 50, 52, 56, 58, 60, 62, 64, 72, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 128, 144, 160, 164, 168, 192, 194, 196, 200, 202, 208, 210, 212, 216, 218, 224, 226, 228

Offset: 1

Robert Israel, Aug 05 2016

nonn

0, and terms of A065609 that are not in A121016.
Number of terms with d binary digits is A001037(d).
Take the binary representation of a(n), reverse it, add 1 to each digit.  The result is the decimal representation of A102659(n).
From Gus Wiseman, Apr 19 2020: (Start)
Also numbers k such that the k-th composition in standard order (row k of A066099) is a Lyndon word. For example, the sequence of all Lyndon words begins:
()            52: (1,2,3)        118: (1,1,2,1,2)
(1)           56: (1,1,4)        120: (1,1,1,4)
(2)           58: (1,1,2,2)      122: (1,1,1,2,2)
(3)           60: (1,1,1,3)      124: (1,1,1,1,3)
(1,2)         62: (1,1,1,1,2)    126: (1,1,1,1,1,2)
(4)           64: (7)            128: (8)
(1,3)         72: (3,4)          144: (3,5)
(1,1,2)       80: (2,5)          160: (2,6)
(5)           84: (2,2,3)        164: (2,3,3)
(2,3)         96: (1,6)          168: (2,2,4)
(1,4)         98: (1,4,2)        192: (1,7)
(1,2,2)      100: (1,3,3)        194: (1,5,2)
(1,1,3)      104: (1,2,4)        196: (1,4,3)
(1,1,1,2)    106: (1,2,2,2)      200: (1,3,4)
(6)          108: (1,2,1,3)      202: (1,3,2,2)
(2,4)        112: (1,1,5)        208: (1,2,5)
(1,5)        114: (1,1,3,2)      210: (1,2,3,2)
(1,3,2)      116: (1,1,2,3)      212: (1,2,2,3)
(End)

			6 is in the sequence because its binary representation 110 is greater than all the rotations 011 and 101.
10 is not in the sequence because its binary representation 1010 is unchanged under rotation by 2 places.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    6:     110 ~ {2,3}
    8:    1000 ~ {4}
   12:    1100 ~ {3,4}
   14:    1110 ~ {2,3,4}
   16:   10000 ~ {5}
   20:   10100 ~ {3,5}
   24:   11000 ~ {4,5}
   26:   11010 ~ {2,4,5}
   28:   11100 ~ {3,4,5}
   30:   11110 ~ {2,3,4,5}
   32:  100000 ~ {6}
   40:  101000 ~ {4,6}
   48:  110000 ~ {5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   56:  111000 ~ {4,5,6}
   58:  111010 ~ {2,4,5,6}
(End)

Robert Israel, Table of n, a(n) for n = 1..9868

A similar concept is A328596.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is a necklace are A328595.
Binary necklaces are A000031.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Length of Lyndon factorization of binary expansion is A211100.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
Cf. A000031, A000740, A008965, A027375, A102659, A121016.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692 (this sequence).
- Co-Lyndon compositions are A326774.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
Cf. A034691, A060223, A124767, A269134, A292884.

filter:= proc(n) local L, k;
  L:= convert(convert(n,binary),string);
  for k from 1 to length(L)-1 do
    if lexorder(L,StringTools:-Rotate(L,k)) then return false fi;
  od;
  true
end proc:
select(filter, [$0..1000]);

filterQ[n_] := Module[{bits, rr}, bits = IntegerDigits[n, 2]; rr = NestList[RotateRight, bits, Length[bits]-1] // Rest; AllTrue[rr, FromDigits[#, 2] < n&]];
Select[Range[0, 1000], filterQ] (* Jean-François Alcover, Apr 29 2019 *)

def ok(n):
    b = bin(n)[2:]
    return all(b[i:] + b[:i] < b for i in range(1, len(b)))
print([k for k in range(230) if ok(k)]) # Michael S. Branicky, May 26 2022

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)

A211100 Number of factors in Lyndon factorization of binary expansion of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 2, 3, 2, 6, 3, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 7, 3, 4, 3, 5, 3, 4, 3, 6, 4, 5, 3, 5, 4, 4, 3, 7, 4, 5, 4, 6, 5, 5, 4, 7

Offset: 0

N. J. A. Sloane, Mar 31 2012

nonn

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). a(n) = number of factors in Lyndon factorization of binary expansion of n.
It appears that a(n) = k for the first time when n = 2^(k-1)+1.
We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. - Gus Wiseman, Nov 12 2019

			n=25 has binary expansion 11001, which has Lyndon factorization (1)(1)(001) with three factors, so a(25) = 3.
Here are the Lyndon factorizations for small values of n:
.0.
.1.
.1.0.
.1.1.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.1.0.0.0.
.1.001.
.1.01.0.
.1.011.
.1.1.0.0.
...

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Maple programs for A211100 etc.

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof).
A211095 and A211096 give information about the smallest (or rightmost) factor. Cf. A211097, A211098, A211099.
Row-lengths of A329314.
The "co-" version is A329312.
Positions of 2's are A329327.
The reversed version is A329313.
The inverted version is A329312.
Ignoring the first digit gives A211097.
Cf. A059966, A060223, A275692, A296658, A328594, A328595, A328596, A329131, A329325, A329326.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Table[Length[lynfac[IntegerDigits[n,2]]],{n,0,30}] (* Gus Wiseman, Nov 12 2019 *)

A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 1, 3, 2, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3

Offset: 0

Gus Wiseman, Nov 11 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

			The sequence of reversed binary expansions of the nonnegative integers together with their Lyndon factorizations begins:
   0:      () = ()
   1:     (1) = (1)
   2:    (01) = (01)
   3:    (11) = (1)(1)
   4:   (001) = (001)
   5:   (101) = (1)(01)
   6:   (011) = (011)
   7:   (111) = (1)(1)(1)
   8:  (0001) = (0001)
   9:  (1001) = (1)(001)
  10:  (0101) = (01)(01)
  11:  (1101) = (1)(1)(01)
  12:  (0011) = (0011)
  13:  (1011) = (1)(011)
  14:  (0111) = (0111)
  15:  (1111) = (1)(1)(1)(1)
  16: (00001) = (00001)
  17: (10001) = (1)(0001)
  18: (01001) = (01)(001)
  19: (11001) = (1)(1)(001)
  20: (00101) = (00101)

The non-reversed version is A211100.
Positions of 1's are A328596.
The "co" version is A329326.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose reversed binary expansion is a aperiodic are A328594.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A027375, A059966, A060223, A121016, A211097, A275692, A329131, A329314,  A329317, A329325.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
Table[If[n==0,0,Length[lynfac[Reverse[IntegerDigits[n,2]]]]],{n,0,30}]

A329318 List of co-Lyndon words on {1,2} sorted first by length and then lexicographically.

Original entry on oeis.org

1, 2, 21, 211, 221, 2111, 2211, 2221, 21111, 21211, 22111, 22121, 22211, 22221, 211111, 212111, 221111, 221121, 221211, 222111, 222121, 222211, 222221, 2111111, 2112111, 2121111, 2121211, 2211111, 2211121, 2211211, 2212111, 2212121, 2212211, 2221111, 2221121

Offset: 1

Gus Wiseman, Nov 11 2019

nonn

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

The non-"co" version is A102659.
Numbers whose binary expansion is co-Lyndon are A275692.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A001037, A027375, A059966, A060223, A211100, A328596, A329324.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Join@@Table[FromDigits/@Select[Tuples[{1,2},n],colynQ],{n,5}]

A211097 Number of factors in Lyndon factorization of binary vectors of lengths 1, 2, 3, ...

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 3, 3, 4, 1, 2, 1, 3, 2, 2, 1, 4, 2, 3, 2, 4, 3, 4, 4, 5, 1, 2, 1, 3, 1, 2, 1, 4, 2, 3, 1, 3, 2, 2, 1, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 2, 3, 2, 4, 3, 2, 1, 4, 2, 3, 2, 3, 2, 2, 1, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5

Offset: 1

N. J. A. Sloane, Apr 01 2012

nonn

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). Here we look at the Lyndon factorizations of the binary vectors 0,1, 00,01,10,11, 000,001,010,011,100,101,110,111, 0000,...
For the largest (or leftmost) factor see A211098, A211099.
The smallest (or rightmost) factors are given by A211095 and A211096, offset by 2.

			Here are the Lyndon factorizations of the first few binary vectors:
.0.
.1.
.0.0.
.01.
.1.0.
.1.1.
.0.0.0.
.001.
.01.0. <- this means that the factorization is (01)(0), for example
.011.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.0.0.0.0.
...

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Maple programs for A211097 etc.

A211098 and A211099 give information about the largest (or leftmost) factor.
Cf. A211095, A211096.
Row-lengths of A329325.
The "co" version is A329400.
Retaining the first digit gives A211100.
Binary Lyndon words are counted by A001037 and constructed by A102659.
Numbers whose reversed binary expansion is Lyndon are A328596.
Cf. A059966, A060223, A275692, A329312, A329313, A329314, A329326.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Table[Length[lynfac[Rest[IntegerDigits[n,2]]]],{n,2,50}] (* Gus Wiseman, Nov 14 2019 *)

A329395 Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 13, 15, 16, 22, 25, 31, 32, 36, 42, 46, 49, 53, 59, 63, 64, 76, 82, 94, 97, 109, 115, 127, 128, 136, 148, 156, 162, 166, 169, 170, 172, 181, 182, 190, 193, 201, 202, 211, 213, 214, 217, 221, 227, 235, 247, 255, 256, 280, 292, 306, 308

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
Conjecture: also numbers k such that the k-th composition in standard order (A066099) is a palindrome, cf. A025065, A242414, A317085, A317086, A317087, A335373. - Gus Wiseman, Jun 06 2020

			The sequence of terms together with their trimmed binary expansions and their co-Lyndon and Lyndon factorizations begins:
   1:      () =               0 = 0
   2:     (0) =             (0) = (0)
   3:     (1) =             (1) = (1)
   4:    (00) =          (0)(0) = (0)(0)
   7:    (11) =          (1)(1) = (1)(1)
   8:   (000) =       (0)(0)(0) = (0)(0)(0)
  10:   (010) =         (0)(10) = (01)(0)
  13:   (101) =         (10)(1) = (1)(01)
  15:   (111) =       (1)(1)(1) = (1)(1)(1)
  16:  (0000) =    (0)(0)(0)(0) = (0)(0)(0)(0)
  22:  (0110) =        (0)(110) = (011)(0)
  25:  (1001) =        (100)(1) = (1)(001)
  31:  (1111) =    (1)(1)(1)(1) = (1)(1)(1)(1)
  32: (00000) = (0)(0)(0)(0)(0) = (0)(0)(0)(0)(0)
  36: (00100) =     (0)(0)(100) = (001)(0)(0)
  42: (01010) =     (0)(10)(10) = (01)(01)(0)
  46: (01110) =       (0)(1110) = (0111)(0)
  49: (10001) =       (1000)(1) = (1)(0001)
  53: (10101) =     (10)(10)(1) = (1)(01)(01)
  59: (11011) =     (110)(1)(1) = (1)(1)(011)
  63: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Cf. A001037, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329394, A329398.
Cf. A329358, A329399, A329400, A329401.

lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Select[Range[100],Length[lynfac[Rest[IntegerDigits[#,2]]]]==Length[colynfac[Rest[IntegerDigits[#,2]]]]&]

cp's OEIS Frontend

A135255 Partial sums of A102659 read as decimal integers.

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A001037 Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.

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A275692 Numbers k such that every rotation of the binary digits of k is less than k.

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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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A211100 Number of factors in Lyndon factorization of binary expansion of n.

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A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

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A329318 List of co-Lyndon words on {1,2} sorted first by length and then lexicographically.

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