A000031 -id:A000031 - 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

A008965 Number of necklaces of sets of beads containing a total of n beads.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 19, 35, 59, 107, 187, 351, 631, 1181, 2191, 4115, 7711, 14601, 27595, 52487, 99879, 190745, 364723, 699251, 1342183, 2581427, 4971067, 9587579, 18512791, 35792567, 69273667, 134219795, 260301175, 505294127, 981706831, 1908881899, 3714566311, 7233642929

Offset: 1

Paul Zimmermann

A necklace of sets of beads is a cycle where each element of the cycle is itself a set of beads, the total size being the total number of beads.
Equivalently, a(n) is the number of cyclic compositions of n. These could also be loosely described as cyclic partitions.
Inverse Mobius transform of A059966. - Jianing Song, Nov 13 2021
This sequence seems to represent the number of modal families (i.e., sets of scales who are each other's modes) in a musical tuning with n notes per octave. - Luke Knotts, May 28 2025

			E.g. the 5 necklaces for n=4 are (3, 1), (4), (1, 1, 1, 1), (2, 1, 1), (2, 2).
In the Combstruct language these can be described as Cycle(Set(Z), Set(Z), Set(Z), Set(Z)), Cycle(Set(Z, Z), Set(Z), Set(Z)), Cycle(Set(Z, Z, Z, Z)), Cycle(Set(Z, Z), Set(Z, Z)), Cycle(Set(Z), Set(Z, Z, Z)).
For n=6 the 13 necklaces are
   1:  (1, 1, 1, 1, 1, 1),
   2:  (2, 1, 1, 1, 1),
   3:  (2, 1, 2, 1),
   4:  (2, 2, 1, 1),
   5:  (2, 2, 2),
   6:  (3, 1, 1, 1),
   7:  (3, 1, 2),
   8:  (3, 2, 1),
   9:  (3, 3),
  10:  (4, 1, 1),
  11:  (4, 2),
  12:  (5, 1),
  13:  (6).
[corrected by Marcel Vonk (mail(AT)marcelvonk.nl), Feb 05 2008]

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 520, Table 8.13.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. Bekes, J. Pedersen and B. Shao, Mad tea party cyclic partitions, College Math. J., 43 (2012), 24-36.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs., Vol. 3 (2000), #00.1.5.
Zuling Chang, Martianus Frederic Ezerman, Adamas Aqsa Fahreza, and Qiang Wang, A Graph Joining Greedy Approach to Binary de Bruijn Sequences, arXiv:2004.09810 [cs.IT], 2020.
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 48.
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), #16.8.2.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021.
Index entries for sequences related to necklaces

Row sums of A037306. CIK transform of A057427.

Cf. A000031.

with(combstruct): seq(combstruct[count]([ N,{N=Cycle(Set(Z,card>=1))},unlabeled ], size=n), n=1..100);

a[n_] := Sum[ EulerPhi[d]*2^(n/d), {d, Divisors[n]}]/n-1; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Sep 04 2012, from A000031 *)
nn=35; Drop[Apply[Plus,Table[CoefficientList[Series[CycleIndex[ CyclicGroup[n],s]/.Table[s[i]->x^i/(1-x^i),{i,1,n}],{x,0,nn}],x],{n,1,nn}]],1]  (* Geoffrey Critzer, Oct 30 2012 *)

N=66;  x='x+O('x^N);
B(x)=x/(1-x);
A=sum(k=1,N,eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
/* Joerg Arndt, Aug 06 2012 */

from sympy import totient, divisors
def A008965(n): return sum(totient(d)*(1<Chai Wah Wu, Sep 23 2023

a(n) = A000031(n) - 1.
G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x)=x/(1-x); see the Flajolet/Soria reference. - Joerg Arndt, Aug 06 2012
From Jianing Song, Nov 13 2021: (Start)
a(n) = Sum_{d|(2^n-1)} phi(d)/ord(2,d), where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d.
Dirichlet g.f.: zeta(s) * (f(s+1)/zeta(s+1) - 1), where f(s) = Sum_{n>=1} 2^n/n^s. (End)

cp's OEIS Frontend

A066656 a(n) = A000031(n) - A001037(n).

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A209970 a(n) = 2^n - A000031(n).

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A100447 Bisection of A000031.

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A054058 Inverse Moebius transform of A000031 (starting at term 0).

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A054079 Moebius transform of A000031 (starting at term 0).

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A100446 Bisection of A000031.

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A322037 a(n) = 2^b(n), where b(n) = A000031(n).

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A054185 Binomial transform of A000031.

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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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A008965 Number of necklaces of sets of beads containing a total of n beads.

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