A027375 -id:A027375 - 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

A000740 Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.

Original entry on oeis.org

1, 1, 3, 6, 15, 27, 63, 120, 252, 495, 1023, 2010, 4095, 8127, 16365, 32640, 65535, 130788, 262143, 523770, 1048509, 2096127, 4194303, 8386440, 16777200, 33550335, 67108608, 134209530, 268435455, 536854005, 1073741823, 2147450880

Offset: 1

N. J. A. Sloane

Also number of compositions of n into relatively prime parts (that is, the gcd of all the parts is 1). Also number of subsets of {1,2,..,n} containing n and consisting of relatively prime numbers. - Vladeta Jovovic, Aug 13 2003
Also number of perfect parity patterns that have exactly n columns (see A118141). - Don Knuth, May 11 2006
a(n) is odd if and only if n is squarefree (Tim Keller). - Emeric Deutsch, Apr 27 2007
a(n) is a multiple of 3 for all n>=3 (see Problem 11161 link). - Emeric Deutsch, Aug 13 2008
Row sums of triangle A143424. - Gary W. Adamson, Aug 14 2008
a(n) is the number of monic irreducible polynomials with nonzero constant coefficient in GF(2)[x] of degree n. - Michel Marcus, Oct 30 2016
a(n) is the number of aperiodic compositions of n, the number of compositions of n with relatively prime parts, and the number of compositions of n with relatively prime run-lengths. - Gus Wiseman, Dec 21 2017

			For n=4, there are 6 compositions of n into coprime parts: <3,1>, <2,1,1>, <1,3>, <1,2,1>, <1,1,2>, and <1,1,1,1>.
From _Gus Wiseman_, Dec 19 2017: (Start)
The a(6) = 27 aperiodic compositions are:
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1122), (1131), (1221), (1311), (2112), (2211), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (24), (42), (51),
  (6).
The a(6) = 27 compositions into relatively prime parts are:
  (111111),
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1122), (1131), (1212), (1221), (1311), (2112), (2121), (2211), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (51).
The a(6) = 27 compositions with relatively prime run-lengths are:
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1131), (1212), (1221), (1311), (2112), (2121), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (24), (42), (51),
  (6).
(End)

H. O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag; contribution by A. Douady, p. 165.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..3322 (terms 1..300 from T. D. Noe)
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014. See Table 2.
Donald Knuth, Robin Chapman and Reiner Martin, Problem 11243, Perfect Parity Patterns, Am. Math. Monthly 115 (7) (2008) p 668, function c(n).
Emeric Deutsch and Lafayette College Problem Group, Problem 11161: Compositions without Common Factors, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363.
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 2.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Nicolae Mihalache and Francois Vigneron, Factorization of the quadratic Misiurewicz-Thurston polynomials, arXiv:2506.17662 [math.DS], 2025. See p. 8.
Robert Munafo, Enumeration of Period-N Mu-Atoms
Jeffrey Shallit and N. J. A. Sloane, Correspondence 1974-1975
François Vigneron and Nicolae Mihalache, How to split a tera-polynomial, arXiv:2402.06083 [math.NA], 2024.
Index entries for sequences related to Lyndon words

Cf. A000837, A003239, A008683, A008965, A022553, A034738, A035928, A038199, A051168, A054525, A056267, A059966, A143424, A167606, A178472, A216954, A228369, A294859, A296302.
Equals A027375/2.
See A056278 for a variant.
First differences of A085945.
Column k=2 of A143325.
Row sums of A101391.

with(numtheory): a[1]:=1: a[2]:=1: for n from 3 to 32 do div:=divisors(n): a[n]:=2^(n-1)-sum(a[n/div[j]],j=2..tau(n)) od: seq(a[n],n=1..32); # Emeric Deutsch, Apr 27 2007
with(numtheory); A000740:=n-> add(mobius(n/d)*2^(d-1), d in divisors(n)); # N. J. A. Sloane, Oct 18 2012

a[n_] := Sum[ MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 03 2012, after PARI *)

a(n) = sumdiv(n,d,moebius(n/d)*2^(d-1))

from sympy import mobius, divisors
def a(n): return sum([mobius(n // d) * 2**(d - 1) for d in divisors(n)])
[a(n) for n in range(1, 101)]  # Indranil Ghosh, Jun 28 2017

a(n) = Sum_{d|n} mu(n/d)*2^(d-1), Mobius transform of A011782. Furthermore, Sum_{d|n} a(d) = 2^(n-1).
a(n) = A027375(n)/2 = A038199(n)/2.
a(n) = Sum_{k=0..n} A051168(n,k)*k. - Max Alekseyev, Apr 09 2013
Recurrence relation: a(n) = 2^(n-1) - Sum_{d|n,d>1} a(n/d). (Lafayette College Problem Group; see the Maple program and Iglesias eq (6)). - Emeric Deutsch, Apr 27 2007
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 2*x^k). - Ilya Gutkovskiy, Oct 24 2018
G.f. satisfies Sum_{n>=1} A( (x/(1 + 2*x))^n ) = x. - Paul D. Hanna, Apr 02 2025

Connection with Mandelbrot set discovered by Warren D. Smith and proved by Robert Munafo, Feb 06 2000

Ambiguous term a(0) removed by Max Alekseyev, Jan 02 2012

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 14, 20, 36, 60, 108, 188, 352, 632, 1182, 2192, 4116, 7712, 14602, 27596, 52488, 99880, 190746, 364724, 699252, 1342184, 2581428, 4971068, 9587580, 18512792, 35792568, 69273668, 134219796, 260301176, 505294128, 981706832

Offset: 0

N. J. A. Sloane

Also a(n)-1 is the number of 1's in the truth table for the lexicographically least de Bruijn cycle (Fredricksen).
In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system. - Paul Cantrell, Dec 28 2011
Also, minimum cardinality of an unavoidable set of length-n binary words (Champarnaud, Hansel, Perrin). - Jeffrey Shallit, Jan 10 2019
(1/n) * Dirichlet convolution of phi(n) and 2^n, n>0. - Richard L. Ollerton, May 06 2021
From Jianing Song, Nov 13 2021: (Start)
a(n) is even for n != 0, 2. Proof: write n = 2^e * s with odd s, then a(n) * s = Sum_{d|s} Sum_{k=0..e} phi((2^e*s)/(2^k*d)) * 2^(2^k*d-e) = Sum_{d|s} Sum_{k=0..e-1} phi(s/d) * 2^(2^k*d-k-1) + Sum_{d|s} phi(s/d) * 2^(2^e*d-e) == Sum_{k=0..e-1} 2^(2^k*s-k-1) + 2^(2^e*s-e) == Sum_{k=0..min{e-1,1}} 2^(2^k*s-k-1) (mod 2). a(n) is odd if and only if s = 1 and e-1 = 0, or n = 2.
a(n) == 2 (mod 4) if and only if n = 1, 4 or n = 2*p^e with prime p == 3 (mod 4).
a(n) == 4 (mod 8) if and only if n = 2^e, 3*2^e for e >= 3, or n = p^e, 4*p^e != 12 with prime p == 3 (mod 4), or n = 2s where s is an odd number such that phi(s) == 4 (mod 8). (End)

			For n=3 and n=4 the necklaces are {000,001,011,111} and {0000,0001,0011,0101,0111,1111}.
The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111...}.

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, pp. 120, 172.
May, Robert M. "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
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

Seiichi Manyama, Table of n, a(n) for n = 0..3333 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, arXiv:2308.16257 [math.CO], 2023. See references.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151, pp. 379-383.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
J.-M. Champarnaud, G. Hansel, and D. Perrin, Unavoidable sets of constant length, Internat. J. Alg. Comput. 14 (2004), 241-251.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
S. N. Ethier and J. Lee, Parrondo games with spatial dependence, arXiv preprint arXiv:1202.2609 [math.PR], 2012.
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.4.7.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see pages 18, 64.
H. Fredricksen, The lexicographically least de Bruijn cycle, J. Combin. Theory, 9 (1970) 1-5.
Harold Fredricksen, An algorithm for generating necklaces of beads in two colors, Discrete Mathematics, Volume 61, Issues 2-3, September 1986, Pages 181-188.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 2
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 130
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
Abraham Lempel, On extremal factors of the de Bruijn graph, J. Combinatorial Theory Ser. B 11 1971 17--27. MR0276126 (43 #1874).
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Johannes Mykkeltveit, A proof of Golomb's conjecture for the de Bruijn graph, J. Combinatorial Theory Ser. B 13 (1972), 40-45. MR0323629 (48 #1985).
Matthew Parker, The first 25K terms (7-Zip compressed file) [a large file]
J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Ville Salo, Universal gates with wires in a row, arXiv:1809.08050 [math.GR], 2018.
J. A. Siehler, The Finite Lamplighter Groups: A Guided Tour, College Mathematics Journal, Vol. 43, No. 3 (May 2012), pp. 203-211.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
David Thomson, Musical Polygons, Mathematics Today, Vol. 57, No. 2 (April 2021), pp. 50-51.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012.
Eric Weisstein's World of Mathematics, Necklace
Wolfram Research, Number of necklaces
Index entries for "core" sequences
Index entries for sequences related to necklaces

Column 2 of A075195.
Cf. A001037 (primitive solutions to same problem), A014580, A000016, A000013, A000029 (if turning over is allowed), A000011, A001371, A058766.
Rows sums of triangle in A047996.
Dividing by 2 gives A053634.
A008965(n) = a(n) - 1 allowing different offsets.
Cf. A008965, A053635, A052823, A100447 (bisection).
Cf. A000010.

a000031 0 = 1
a000031 n = (`div` n) $ sum $
   zipWith (*) (map a000010 divs) (map a000079 $ reverse divs)
   where divs = a027750_row n
-- Reinhard Zumkeller, Mar 21 2013

with(numtheory); A000031 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*2^(n/d); od; RETURN(s/n); fi; end; [ seq(A000031(n), n=0..50) ];

a[n_] := Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n/d), 0], {d, 1, n}]/n
a[n_] := Fold[#1 + 2^(n/#2) EulerPhi[#2] &, 0, Divisors[n]]/n (* Ben Branman, Jan 08 2011 *)
Table[Expand[CycleIndex[CyclicGroup[n], t] /. Table[t[i]-> 2, {i, 1, n}]], {n,0, 30}] (* Geoffrey Critzer, Mar 06 2011*)
a[0] = 1; a[n_] := DivisorSum[n, EulerPhi[#]*2^(n/#)&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2016 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-2*x^i]/i,{i,1,mx}],{x,0,mx}],x] (*Herbert Kociemba, Oct 29 2016 *)

{A000031(n)=if(n==0,1,sumdiv(n,d,eulerphi(d)*2^(n/d))/n)} \\ Randall L Rathbun, Jan 11 2002

from sympy import totient, divisors
def A000031(n): return sum(totient(d)*(1<Chai Wah Wu, Nov 16 2022

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d) = A053635(n)/n, where phi is A000010.
Warning: easily confused with A001037, which has a similar formula.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n. - Herbert Kociemba, Oct 29 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 2^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021
Dirichlet g.f.: f(s+1) * (zeta(s)/zeta(s+1)), where f(s) = Sum_{n>=1} 2^n/n^s. - Jianing Song, Nov 13 2021

There is an error in Fig. M3860 in the 1995 Encyclopedia of Integer Sequences: in the third line, the formula for A000031 = M0564 should be (1/n) sum phi(d) 2^(n/d).

A000048 Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 9, 16, 28, 51, 93, 170, 315, 585, 1091, 2048, 3855, 7280, 13797, 26214, 49929, 95325, 182361, 349520, 671088, 1290555, 2485504, 4793490, 9256395, 17895679, 34636833, 67108864, 130150493, 252645135, 490853403, 954437120, 1857283155

Offset: 0

N. J. A. Sloane

Also, for any m which is a multiple of n, the number of 2m-bead balanced binary necklaces of fundamental period 2n that are equivalent to their complements. [Clarified by Aaron Meyerowitz, Jun 01 2024]
Also binary Lyndon words of length n with an odd number of 1's (for n>=1).
Also number of binary irreducible polynomials of degree n having trace 1.
Also number of binary irreducible polynomials of degree n having linear coefficient 1 (this is the same as the trace-1 condition, as the reciprocal of an irreducible polynomial is again irreducible).
Also number of binary irreducible self-reciprocal polynomials of degree 2*n; there is no such polynomial for odd degree except for x+1.
Also number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 1 (mod n+1) = size of Varshamov-Tenengolts code VT_1(n).
Also the number of dynamical cycles of period 2n of a threshold Boolean automata network which is a quasi-minimal negative circuit of size nq where q is odd and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009
Also the number of 3-elements orbits of the symmetric group S3 action on irreducible polynomials of degree 2n, n>1, over GF(2). - Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Oct 04 2009
Conjecture: Also the number of caliber-n cycles of Zagier-reduced indefinite binary quadratic forms with sum invariant equal to s, where (s-1)/n is an odd integer. - Barry R. Smith, Dec 14 2014
The Metropolis, Stein, Stein (1973) reference on page 31 Table II lists a(k) for k = 2 to 15 and is actually for sequence A056303 since there a(k) = 0 for k<2. - Michael Somos, Dec 20 2014

			a(5) = 3 corresponding to the necklaces 00001, 00111, 01011.
a(6) = 5 from 000001, 000011, 000101, 000111, 001011.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
H. Kawakami, Table of rotation sequences of x_{n+1} = x_n^2 - lambda, pp. 73-92 of G. Ikegami, Editor, Dynamical Systems and Nonlinear Oscillations, Vol. 1, World Scientific, 1986.
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
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..3320 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), p.408 and p.848.
L. Carlitz, A theorem of Dickson on irreducible polynomials, Proc. Amer. Math. Soc. 3, (1952). 693-700.
CombOS - Combinatorial Object Server, Generate necklaces, Lyndon words, chord diagrams, and relatives
J. Demongeot, M. Noual and S. Sene, On the number of attractors of positive and negative threshold Boolean automata circuits, hal-00647877 preprint (2009). [From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009]
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 2.
J. E. Iglesias, Enumeration of polytypes MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 178-194, Table 1.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.
N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44; reprinted in P. Cvitanovic, ed., Universality in Chaos, Hilger, Bristol, 1986, pp. 187-206.
H. Meyn and W. Götz, Self-reciprocal polynomials over finite fields, Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
Simon Michalowsky, Bahman Gharesifard, Christian Ebenbauer, A Lie bracket approximation approach to distributed optimization over directed graphs, arXiv:1711.05486 [math.OC], 2017.
Tilman Piesk, Lists of the three sets of necklaces for n=1..12
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
Frank Ruskey, Number of q-ary Lyndon words with given trace mod q
Frank Ruskey, Number of Lyndon words over GF(q) with given trace
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane, On single-deletion-correcting codes
B. R. Smith, Reducing quadratic forms by kneading sequences, Journal of Integer Sequences, 17 (2014), article 14.11.8.
P. R. Stein, Letter to N. J. A. Sloane, Jun 02 1971
J.-Y. Thibon, The cycle enumerator of unimodal permutations, arXiv:math/0102051 [math.CO], 2001.
Index entries for "core" sequences
Index entries for sequences related to Lyndon words
Index entries for sequences related to subset sums modulo m

Like A000013, but primitive necklaces. Half of A064355.
Equals A042981 + A042982.
Cf. A002823, A000016, A053633, A051841, A001037, A002075, A002076, A008683.
Cf. also A001037, A056303.
Very close to A006788 [Fisher, 1989].
bisection (odd terms) is A131203

with(numtheory); A000048 := proc(n) local d,t1; if n = 0 then RETURN(1) else t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+mobius(d)*2^(n/d)/(2*n); fi; od; RETURN(t1); fi; end;

a[n_] := Total[ MoebiusMu[#]*2^(n/#)& /@ Select[ Divisors[n], OddQ]]/(2n); a[0] = 1; Table[a[n], {n,0,35}] (* Jean-François Alcover, Jul 21 2011 *)
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, MoebiusMu[#] 2^(n/#) &, OddQ] / (2 n)]; (* Michael Somos, Dec 20 2014 *)

A000048(n) = sumdiv(n,d,(d%2)*(moebius(d)*2^(n/d)))/(2*n) \\ Michael B. Porter, Nov 09 2009

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%2==1, L(n, k), 0 ) ) / n;
vector(55,n,a(n)) \\ Joerg Arndt, Jun 28 2012

from sympy import divisors, mobius
def a(n): return 1 if n<1 else sum(mobius(d)*2**(n//d) for d in divisors(n) if d%2)//(2*n) # Indranil Ghosh, Apr 28 2017

a(n) = (1/(2*n)) * Sum_{odd d divides n} mu(d)*2^(n/d), where mu is the Mobius function A008683.
a(n) = A056303(n) for all integer n>=2. - Michael Somos, Dec 20 2014
Sum_{k dividing m for which m/k is odd} k*a(k) = 2^(m-1). (This explains the observation that the sequence is very close to A006788. Unless m has some nontrivial odd divisors that are small relative to m, the term m*a(m) will dominate the sum. Thus, we see for instance that a(n) = A006788(n) when n has one of the forms 2^m or 2^m*p where p is an odd prime with a(2^m) < p.) - Barry R. Smith, Oct 24 2015
A000013(n) = Sum_{d|n} a(d). - Robert A. Russell, Jun 09 2019
G.f.: 1 + Sum_{k>=1} mu(2*k)*log(1 - 2*x^k)/(2*k). - Ilya Gutkovskiy, Nov 11 2019

Additional comments from Frank Ruskey, Dec 13 1999

A328596 Numbers whose reversed binary expansion is a Lyndon word (aperiodic necklace).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 26, 28, 30, 32, 40, 44, 48, 52, 56, 58, 60, 62, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 128, 144, 152, 160, 164, 168, 172, 176, 180, 184, 188, 192, 200, 208, 212, 216, 218, 220, 224

Offset: 1

Gus Wiseman, Oct 22 2019

First differs from A091065 in lacking 50.

A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.

			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}
  44: 101100 ~ {3,4,6}
  48: 110000 ~ {5,6}
  52: 110100 ~ {3,5,6}
  56: 111000 ~ {4,5,6}
  58: 111010 ~ {2,4,5,6}

A similar concept is A275692.
Aperiodic words are A328594.
Necklaces are A328595.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Cf. A000031, A000120, A000740, A008965, A027375, A121016.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],aperQ[Reverse[IntegerDigits[#,2]]]&&neckQ[Reverse[IntegerDigits[#,2]]]&]

Intersection of A328594 and A328595.

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

A328594 Numbers whose binary expansion is aperiodic.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Oct 22 2019

nonn

A finite sequence is aperiodic if all of its cyclic rotations are distinct. See A000740 or A027375 for details.

Also numbers k such that the k-th composition in standard order is aperiodic. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 28 2020

			The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}

The complement is A121016.
The version for prime indices is A085971.
Numbers without proper integer roots are A007916.
Necklaces are A328595.
Lyndon words are A328596.
Aperiodic compositions are A000740.
Aperiodic binary sequences are A027375.
Cf. A000120, A000939, A014081, A065609, A069010, A275692, A323867, A334030.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Select[Range[0,100],aperQ[IntegerDigits[#,2]]&]

A178472 Number of compositions (ordered partitions) of n where the gcd of the part sizes is not 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 1, 32768, 1027, 65537, 79, 133088, 1, 262145, 4099, 524408, 1, 1056731, 1, 2097158, 16636, 4194305, 1, 8421248, 64, 16777712, 65539

Offset: 1

Franklin T. Adams-Watters, May 28 2010

nonn

Of course, all part sizes must be greater than 1; that condition alone gives the Fibonacci numbers, which is thus an upper bound.

Also the number of periodic compositions of n, where a sequence is periodic if its cyclic rotations are not all different. Also compositions with non-relatively prime run-lengths. - Gus Wiseman, Nov 10 2019

			For n=6, we have 5 compositions: <6>, <4,2>, <2,4>, <2,2,2>, and <3,3>.
From _Gus Wiseman_, Nov 10 2019: (Start)
The a(2) = 1 through a(9) = 4 non-relatively prime compositions:
  (2)  (3)  (4)    (5)  (6)      (7)  (8)        (9)
            (2,2)       (2,4)         (2,6)      (3,6)
                        (3,3)         (4,4)      (6,3)
                        (4,2)         (6,2)      (3,3,3)
                        (2,2,2)       (2,2,4)
                                      (2,4,2)
                                      (4,2,2)
                                      (2,2,2,2)
The a(2) = 1 through a(9) = 4 periodic compositions:
  11  111  22    11111  33      1111111  44        333
           1111         222              1313      121212
                        1212             2222      212121
                        2121             3131      111111111
                        111111           112112
                                         121121
                                         211211
                                         11111111
The a(2) = 1 through a(9) = 4 compositions with non-relatively prime run-lengths:
  11  111  22    11111  33      1111111  44        333
           1111         222              1133      111222
                        1122             2222      222111
                        2211             3311      111111111
                        111111           111122
                                         112211
                                         221111
                                         11111111
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014. See Table 2.

Cf. A000045, A008683, A011782, A178470.
Periodic binary words are A152061.
Cf. A000740, A027375, A059966, A121016, A329140, A329145.

A178472 := n -> (2^n - add(mobius(n/d)*2^d, d in divisors(n)))/2:
seq(A178472(n), n=1..51); # Peter Luschny, Jan 21 2018

Table[2^(n - 1) - DivisorSum[n, MoebiusMu[n/#]*2^(# - 1) &], {n, 51}] (* Michael De Vlieger, Jan 20 2018 *)

vector(60,n,2^(n-1)-sumdiv(n,d,2^(d-1)*moebius(n/d)))

from sympy import mobius, divisors
def A178472(n): return -sum(mobius(n//d)<Chai Wah Wu, Sep 21 2024

a(n) = Sum_{d|n & d

a(n) = 2^(n-1) - A000740(n).

a(n) = A152061(n)/2. - George Beck, Jan 20 2018

a(p) = 1 for p prime. - Chai Wah Wu, Sep 21 2024

Ambiguous term a(0) removed by Max Alekseyev, Jan 02 2012

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

Showing 1-10 of 84 results. Next

cp's OEIS Frontend

A351839 Triangle read by rows: T(n, k) = A027375(n)*Sum_{m=1..floor(n/k)} binomial(n, k*m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

Extensions

A000740 Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A000048 Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

A351839 Triangle read by rows: T(n, k) = A027375(n)Sum_{m=1..floor(n/k)} binomial(n, km).