A001037 -id:A001037 - cp's OEIS frontend

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.

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).

A059966 a(n) = (1/n) * Sum_{ d divides n } mu(n/d) * (2^d - 1).

Original entry on oeis.org

1, 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

Offset: 1

Roland Bacher, Mar 05 2001

Dimensions of the homogeneous parts of the free Lie algebra with one generator in 1,2,3, etc. (Lie analog of the partition numbers).
This sequence is the Lie analog of the partition sequence (which gives the dimensions of the homogeneous polynomials with one generator in each degree) or similarly, of the partitions into distinct (or odd numbers) (which gives the dimensions of the homogeneous parts of the exterior algebra with one generator in each dimension).
The number of cycles of length n of rectangle shapes in the process of repeatedly cutting a square off the end of the rectangle. For example, the one cycle of length 1 is the golden rectangle. - David Pasino (davepasino(AT)yahoo.com), Jan 29 2009
In music, the number of distinct rhythms, at a given tempo, produced by a continuous repetition of measures with identical patterns of 1's and 0's (where 0 means no beat, and 1 means one beat), where each measure allows for n possible beats of uniform character, and when counted under these two conditions: (i) the starting and ending times for the measure are unknown or irrelevant and (ii) identical rhythms that can be produced by using a measure with fewer than n possible beats are excluded from the count. - Richard R. Forberg, Apr 22 2013
Richard R. Forberg's comment does not hold for n=1 because a(1)=1 but there are the two possible rhythms: "0" and "1". - Herbert Kociemba, Oct 24 2016
The comment does hold for n=1 as the rhythm "0" can be produced by using a measure of 0 beats and so is properly excluded from a(1)=1 by condition (ii) of the comment. - Travis Scott, May 28 2022
a(n) is also the number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n. - Gus Wiseman, Dec 19 2017
Mobius transform of A008965. - Jianing Song, Nov 13 2021
a(n) is the number of cycles of length n for the map x->1 - abs(2*x-1) applied on rationals 0Michel Marcus, Jul 16 2025

			a(4)=3: the 3 elements [a,c], [a[a,b]] and d form a basis of all homogeneous elements of degree 4 in the free Lie algebra with generators a of degree 1, b of degree 2, c of degree 3 and d of degree 4.
From _Gus Wiseman_, Dec 19 2017: (Start)
The sequence of Lyndon compositions organized by sum begins:
  (1),
  (2),
  (3),(12),
  (4),(13),(112),
  (5),(14),(23),(113),(122),(1112),
  (6),(15),(24),(114),(132),(123),(1113),(1122),(11112),
  (7),(16),(25),(115),(34),(142),(124),(1114),(133),(223),(1213),(1132),(1123),(11113),(1222),(11212),(11122),(111112). (End)

C. Reutenauer, Free Lie algebras, Clarendon press, Oxford (1993).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.
S. V. Duzhin and D. V. Pasechnik, Groups acting on necklaces and sandpile groups, Journal of Mathematical Sciences, August 2014, Volume 200, Issue 6, pp 690-697. See page 85. - N. J. A. Sloane, Jun 30 2014
Seok-Jin Kang and Myung-Hwan Kim, Free Lie Algebras, Generalized Witt Formula and the Denominator Identity, Journal of Algebra 183, 560-594 (1996).
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Jakob Oesinghaus, Quasi-symmetric functions and the Chow ring of the stack of expanded pairs, arXiv:1806.10700 [math.AG], 2018.
Robert Schneider, Andrew V. Sills, and Hunter Waldron, On the q-factorization of power series, arXiv:2501.18744 [math.CO], 2025. See p. 6.

Apart from initial terms, same as A001037.

Cf. A000225, A000740, A008683, A008965, A011782, A060223, A185700, A228369, A269134 A281013, A296302, A296373.

a059966 n = sum (map (\x -> a008683 (n `div` x) * a000225 x)
                     [d | d <- [1..n], mod n d == 0]) `div` n
-- Reinhard Zumkeller, Nov 18 2011

Table[1/n Apply[Plus, Map[(MoebiusMu[n/# ](2^# - 1)) &, Divisors[n]]], {n, 20}]
(* Second program: *)
Table[(1/n) DivisorSum[n, MoebiusMu[n/#] (2^# - 1) &], {n, 35}] (* Michael De Vlieger, Jul 22 2019 *)

from sympy import mobius, divisors
def A059966(n): return sum(mobius(n//d)*(2**d-1) for d in divisors(n,generator=True))//n # Chai Wah Wu, Feb 03 2022

G.f.: Product_{n>0} (1-q^n)^a(n) = 1-q-q^2-q^3-q^4-... = 2-1/(1-q).
Inverse Euler transform of A011782. - Alois P. Heinz, Jun 23 2018
G.f.: Sum_{k>=1} mu(k)*log((1 - x^k)/(1 - 2*x^k))/k. - Ilya Gutkovskiy, May 19 2019
a(n) ~ 2^n / n. - Vaclav Kotesovec, Aug 10 2019
Dirichlet g.f.: f(s+1)/zeta(s+1) - 1, where f(s) = Sum_{n>=1} 2^n/n^s. - Jianing Song, Nov 13 2021

Explicit formula from Paul D. Hanna, Apr 15 2002

Description corrected by Axel Kleinschmidt, Sep 15 2002

A006918 a(n) = binomial(n+3, 3)/4 for odd n, n(n+2)(n+4)/24 for even n.

Original entry on oeis.org

0, 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, 112, 140, 168, 204, 240, 285, 330, 385, 440, 506, 572, 650, 728, 819, 910, 1015, 1120, 1240, 1360, 1496, 1632, 1785, 1938, 2109, 2280, 2470, 2660, 2870, 3080, 3311, 3542, 3795, 4048, 4324, 4600, 4900, 5200, 5525, 5850, 6201, 6552, 6930

Offset: 0

N. J. A. Sloane

Maximal number of inconsistent triples in a tournament on n+2 nodes [Kac]. - corrected by Leen Droogendijk, Nov 10 2014
a(n-4) is the number of aperiodic necklaces (Lyndon words) with 4 black beads and n-4 white beads.
a(n-3) is the maximum number of squares that can be formed from n lines, for n>=3. - Erich Friedman; corrected by Leen Droogendijk, Nov 10 2014
Number of trees with diameter 4 where at most 2 vertices 1 away from the graph center have degree > 2. - Jon Perry, Jul 11 2003
a(n+1) is the number of partitions of n into parts of two kinds, with at most two parts of each kind. Also a(n-3) is the number of partitions of n with Durfee square of size 2. - Franklin T. Adams-Watters, Jan 27 2006
Factoring the g.f. as x/(1-x)^2 times 1/(1-x^2)^2 we find that the sequence equals (1, 2, 3, 4, ...) convolved with (1, 0, 2, 0, 3, 0, 4, ...), A000027 convolved with its aerated variant. -  Gary W. Adamson, May 01 2009
Starting with "1" = triangle A171238 * [1,2,3,...]. - Gary W. Adamson, Dec 05 2009
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of this sequence, e.g., Kn22(n) = a(n+1) + a(n) + 2*a(n-1) + a(n-2) and Fi2(n) = a(n) + 4*a(n-1) + a(n-2). - Johannes W. Meijer, May 20 2011
For n>3, a(n-4) is the number of (w,x,y,z) having all terms in {1,...,n} and w+x+y+z=|x-y|+|y-z|. - Clark Kimberling, May 23 2012
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w+x+y < |w-x|+|x-y|. - Clark Kimberling, Jun 13 2012
For n>0 number of inequivalent (n-1) X 2 binary matrices, where equivalence means permutations of rows or columns or the symbol set. - Alois P. Heinz, Aug 17 2014
Number of partitions p of n+5 such that p[3] = 2. Examples: a(1)=1 because we have (2,2,2); a(2)=2 because we have (2,2,2,1) and (3,2,2); a(3)=5 because we have (2,2,2,1,1), (2,2,2,2), (3,2,2,1), (3,3,2), and (4,2,2). See the R. P. Stanley reference. - Emeric Deutsch, Oct 28 2014
Sum over each antidiagonal of A243866. - Christopher Hunt Gribble, Apr 02 2015
Number of nonisomorphic outer planar graphs of order n>=3, size n+2, and maximum degree 3. - Christian Barrientos and Sarah Minion, Feb 27 2018
a(n) is the number of 2413-avoiding odd Grassmannian permutations of size n+1. - Juan B. Gil, Mar 09 2023

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 14*x^5 + 20*x^6 + 30*x^7 + 40*x^8 + 55*x^9 + ...
From _Gus Wiseman_, Apr 06 2019: (Start)
The a(4 - 3) = 1 through a(8 - 3) = 14 integer partitions with Durfee square of length 2 are the following (see Franklin T. Adams-Watters's second comment). The Heinz numbers of these partitions are given by A325164.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(0 + 1) = 1 through a(4 + 1) = 14 integer partitions of n into parts of two kinds with at most two parts of each kind are the following (see Franklin T. Adams-Watters's first comment).
  ()()  ()(1)  ()(2)   ()(3)    ()(4)
        (1)()  (2)()   (3)()    (4)()
               ()(11)  (1)(2)   (1)(3)
               (1)(1)  ()(21)   ()(22)
               (11)()  (2)(1)   (2)(2)
                       (21)()   (22)()
                       (1)(11)  ()(31)
                       (11)(1)  (3)(1)
                                (31)()
                                (11)(2)
                                (1)(21)
                                (2)(11)
                                (21)(1)
                                (11)(11)
The a(6 - 5) = 1 through a(10 - 5) = 14 integer partitions whose third part is 2 are the following (see Emeric Deutsch's comment). The Heinz numbers of these partitions are given by A307373.
  (222)  (322)   (332)    (432)     (442)
         (2221)  (422)    (522)     (532)
                 (2222)   (3222)    (622)
                 (3221)   (3321)    (3322)
                 (22211)  (4221)    (4222)
                          (22221)   (4321)
                          (32211)   (5221)
                          (222111)  (22222)
                                    (32221)
                                    (33211)
                                    (42211)
                                    (222211)
                                    (322111)
                                    (2221111)
(End)

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.
M. Kac, An example of "counting without counting", Philips Res. Reports, 30 (1975), 20*-22* [Special issue in honour of C. J. Bouwkamp].
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.
K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 186, Theorem 6.11.
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. 1, 2nd ed., 2012, Exercise 4.16, pp. 530, 552.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.

T. D. Noe, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Yuriy Choliy and Andrew V. Sills, A formula for the partition function that “counts”, Preprint 2015.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
B. G. Eke, Monotonic triads, Discrete Math., Vol. 9, No. 4 (1974), pp. 359-363. MR0354390 (50 #6869)
Irene Erazo, John López, and Carlos Trujillo, A combinatorial problem that arose in integer B_3 Sets, Revista Colombiana de Matemáticas, Vol. 53, No. 2 (2019), pp. 195-203.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
John Golden and Marcus Spradlin, Collinear and Soft Limits of Multi-Loop Integrands in N= 4 Yang-Mills, arXiv preprint arXiv:1203.1915 [hep-th], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 11.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=2]
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
Index entries for sequences related to Lyndon words.

Cf. A000031, A001037, A028723, A051168. a(n) = T(n,4), array T as in A051168.
Cf. A000094.
Cf. A171238. - Gary W. Adamson, Dec 05 2009
Row sums of A173997. - Gary W. Adamson, Mar 05 2010
Column k=2 of A242093. Column k=2 of A115720 and A115994.
Cf. A117485, A257990, A307373, A325164, A325168.

a006918 n = a006918_list !! n
a006918_list = scanl (+) 0 a008805_list
-- Reinhard Zumkeller, Feb 01 2013

[Floor(Binomial(n+4, 4)/(n+4))-Floor((n+2)/8)*(1+(-1)^n)/2: n in [0..60]]; // Vincenzo Librandi, Nov 10 2014

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=r),U=Sequence(Z,card>=3)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m),m=11..58) ; # Zerinvary Lajos, Mar 09 2007
A006918 := proc(n)
    if type(n,'even') then
        n*(n+2)*(n+4)/24 ;
    else
        binomial(n+3,3)/4 ;
    fi ;
end proc: # R. J. Mathar, May 17 2016

f[n_]:=If[EvenQ[n],(n(n+2)(n+4))/24,Binomial[n+3,3]/4]; Join[{0},Array[f,60]]  (* Harvey P. Dale, Apr 20 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==2&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

{ parttrees(n)=local(pt,k,nk); if (n%2==0, pt=(n/2+1)^2, pt=ceil(n/2)*(ceil(n/2)+1)); pt+=floor(n/2); for (x=1,floor(n/2),pt+=floor(x/2)+floor((n-x)/2)); if (n%2==0 && n>2, pt-=floor(n/4)); k=1; while (3*k<=n, for (x=k,floor((n-k)/2), pt+=floor(k/2); if (x!=k, pt+=floor(x/2)); if ((n-x-k)!=k && (n-x-k)!=x, pt+=floor((n-x-k)/2))); k++); pt }

{a(n) = n += 2; (n^3 - n * (2-n%2)^2) / 24}; /* Michael Somos, Aug 15 2009 */

G.f.: x/((1-x)^2*(1-x^2)^2) = x/((1+x)^2*(1-x)^4).
0, 0, 0, 1, 2, 5, 8, 14, ... has a(n) = (Sum_{k=0..n} floor(k(n-k)/2))/2. - Paul Barry, Sep 14 2003
0, 0, 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, 55, ... has a(n) = binomial(floor(1/2 n), 3) + binomial(floor(1/2 n + 1/2), 3) [Eke]. - N. J. A. Sloane, May 12 2012
a(0)=0, a(1)=1, a(n) = (2/(n-1))*a(n-1) + ((n+3)/(n-1))*a(n-2). - Benoit Cloitre, Jun 28 2004
a(n) = floor(binomial(n+4, 4)/(n+4)) - floor((n+2)/8)(1+(-1)^n)/2. - Paul Barry, Jan 01 2005
a(n+1) = a(n) + binomial(floor(n/2)+2,2), i.e., first differences are A008805. Convolution of A008619 with itself, then shifted right (or A004526 with itself, shifted left by 3). - Franklin T. Adams-Watters, Jan 27 2006
a(n+1) = (A027656(n) + A003451(n+5))/2 with a(1)=0. - Yosu Yurramendi, Sep 12 2008
Linear recurrence: a(n) = 2a(n-1) + a(n-2) - 4a(n-3) + a(n-4) + 2a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008
Euler transform of length 2 sequence [2, 2]. - Michael Somos, Aug 15 2009
a(n) = -a(-4-n) for all n in Z.
a(n+1) + a(n) = A002623(n). - Johannes W. Meijer, May 20 2011
a(n) = (n+2)*(2*n*(n+4)-3*(-1)^n+3)/48. - Bruno Berselli, May 21 2011
a(2n) = A007290(n+2). - Jon Perry, Nov 10 2014
G.f.: (1/(1-x)^4-1/(1-x^2)^2)/4. - Herbert Kociemba, Oct 23 2016
E.g.f.: (x*(18 + 9*x + x^2)*cosh(x) + (6 + 15*x + 9*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, Dec 07 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 75/4 - 24*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 69/4 - 24*log(2). (End)

A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 25, 31, 37, 41, 47, 55, 59, 61, 67, 73, 87, 91, 97, 103, 109, 115, 117, 131, 137, 143, 145, 157, 167, 171, 185, 191, 193, 203, 211, 213, 229, 239, 241, 247, 253, 283, 285, 299, 301, 313, 319, 333, 351, 355, 357, 361, 369, 375

Offset: 1

David Petry (petry(AT)accessone.com)

nonn

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014
2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - Jianing Song, May 10 2021
For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - Joerg Arndt and Jianing Song, May 10 2021

			x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1377 terms from T. D. Noe)
Index entries for sequences operating on GF(2)[X]-polynomials

Written in binary: A058943.
Number of degree-n irreducible polynomials: A001037, see also A000031.
Multiplication table: A048720.
Characteristic function: A091225. Inverse: A091227. a(n) = A091202(A000040(n)). Almost complement of A091242. Union of A091206 & A091214 and also of A091250 & A091252. First differences: A091223. Apart from a(1) and a(2), a subsequence of A092246 and hence A000069.
Table of irreducible factors of n: A256170.
Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570.
Factorization sentinel: A278239.
Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042.
See A115871 for sequences related to cross-domain congruences.
Functions based on the irreducibles: A305421, A305422.

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* Robert G. Wilson v, Aug 12 2011 *)
Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus -> 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* Jean-François Alcover, Nov 21 2016 *)

is(n)=polisirreducible(Pol(binary(n))*Mod(1,2)) \\ Charles R Greathouse IV, Mar 22 2013

A027375 Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n.

Original entry on oeis.org

0, 2, 2, 6, 12, 30, 54, 126, 240, 504, 990, 2046, 4020, 8190, 16254, 32730, 65280, 131070, 261576, 524286, 1047540, 2097018, 4192254, 8388606, 16772880, 33554400, 67100670, 134217216, 268419060, 536870910, 1073708010, 2147483646, 4294901760

Offset: 0

N. J. A. Sloane

A sequence S is aperiodic if it is not of the form S = T^k with k>1. - N. J. A. Sloane, Oct 26 2012
Equivalently, number of output sequences with primitive period n from a simple cycling shift register. - Frank Ruskey, Jan 17 2000
Also, the number of nonempty subsets A of the set of the integers 1 to n such that gcd(A) is relatively prime to n (for n>1). - R. J. Mathar, Aug 13 2006; range corrected by Geoffrey Critzer, Dec 07 2014
Without the first term, this sequence is the Moebius transform of 2^n (n>0). For n > 0, a(n) is also the number of periodic points of period n of the transform associated to the Kolakoski sequence A000002. This 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 2 periodic points of period 2. A001037(n) = a(n)/n gives the number of orbits of size n. - Jean-Christophe Hervé, Oct 25 2014
From Bernard Schott, Jun 19 2019: (Start)
There are 2^n strings of length n that can be formed from the symbols 0 and 1; in the example below with a(3) = 6, the last two strings that are not aperiodic binary strings are { 000, 111 }, corresponding to 0^3 and 1^3, using the notation of the first comment.
Two properties mentioned by Krusemeyer et al. are:
1) For any n > 2, a(n) is divisible by 6.
2) Lim_{n->oo} a(n+1)/a(n) = 2. (End)

			a(3) = 6 = |{ 001, 010, 011, 100, 101, 110 }|. - corrected by _Geoffrey Critzer_, Dec 07 2014

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 13. - From N. J. A. Sloane, Oct 26 2012
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 164.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
Mark I. Krusemeyer, George T. Gilbert, Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 128, pp. 225-227.

T. D. Noe, Table of n, a(n) for n = 0..300
J.-P. Allouche, Note on the transcendence of a generating function. In A. Laurincikas and E. Manstavicius, editors, Proceedings of the Palanga Conference for the 75th birthday of Prof. Kubilius, New trends in Probab. and Statist., Vol. 4, pages 461-465, 1997.
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102 [math.CO], Dec 25 2012.
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
John D. Cook, Counting primitive bit strings (2014).
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 85.
Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit and Taylor J. Smith, Periodicity in rectangular arrays, arXiv:1602.06915 [cs.DM], 2016; Information Processing Letters 118 (2017) 58-63. See Table 1.
O. Georgiou, C. P. Dettmann and E. G. Altmann, Faster than expected escape for a class of fully chaotic maps, arXiv preprint arXiv:1207.7000 [nlin.CD], 2012. - From _N. J. A. Sloane_, Dec 23 2012
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
David W. Lyons, Cristina Mullican, Adam Rilatt, and Jack D. Putnam, Werner states from diagrams, arXiv:2302.05572 [quant-ph], 2023.
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
M. B. Nathanson, Primitive sets and Euler phi function for subsets of {1,2,...,n}, arXiv:math/0608150 [math.NT], 2006-2007.
P. Pongsriiam, Relatively Prime Sets, Divisor Sums, and Partial Sums, arXiv:1306.4891 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.9.1.
P. Pongsriiam, A remark on relative prime sets, arXiv:1306.2529 [math.NT], 2013.
P. Pongsriiam, A remark on relative prime sets, Integers 13 (2013), A49.
R. C. Read, Combinatorial problems in the theory of music, Disc. Math. 167/168 (1997) 543-551, sequence A(n).
M. Tang, Relatively Prime Sets and a Phi Function for Subsets of {1, 2, ... , n}, J. Int. Seq. 13 (2010) # 10.7.6.
László Tóth, Menon-type identities concerning subsets of the set {1,2,...,n}, arXiv:2109.06541 [math.NT], 2021.

A038199 and A056267 are essentially the same sequence with different initial terms.
Cf. A020921, A216953.
Column k=2 of A143324.

a027375 n = n * a001037 n  -- Reinhard Zumkeller, Feb 01 2013

with(numtheory): A027375 :=n->add( mobius(d)*2^(n/d), d = divisors(n)); # N. J. A. Sloane, Sep 25 2012

Table[ Apply[ Plus, MoebiusMu[ n / Divisors[n] ]*2^Divisors[n] ], {n, 1, 32} ]
a[0]=0; a[n_] := DivisorSum[n, MoebiusMu[n/#]*2^#&]; Array[a, 40, 0] (* Jean-François Alcover, Dec 01 2015 *)

PARI
```
a(n) = sumdiv(n,d,moebius(n\d)*2^d);
```

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

a(n) = Sum_{d|n} mu(d)*2^(n/d).
a(n) = 2*A000740(n).
a(n) = n*A001037(n).
Sum_{d|n} a(n) = 2^n.
a(p) = 2^p - 2 for p prime. - R. J. Mathar, Aug 13 2006
a(n) = 2^n - O(2^(n/2)). - Charles R Greathouse IV, Apr 28 2016
a(n) = 2^n - A152061(n). - Bernard Schott, Jun 20 2019
G.f.: 2 * Sum_{k>=1} mu(k)*x^k/(1 - 2*x^k). - Ilya Gutkovskiy, Nov 11 2019

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

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590

Offset: 0

N. J. A. Sloane

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
Number of triangular partitions (see Almkvist).
Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.
  .....{*}..................{*}*.*{*}{*}
  .....*.*....................*.*.*.{*}
  ....*.*.*....---------\......*.*.*
  ..{*}*.*.*...---------/.......*.*
  {*}{*}*.*{*}..................{*}
  - Lekraj Beedassy, Oct 13 2003
Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004
Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004
Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005
Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008
In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008
Column sums of:
1 2 3 4 5 6 7  8  9.....
      1 2 3 4  5  6.....
            1  2  3.....
........................
----------------------
1 2 3 5 7 9 12 15 18  - Jon Perry, Nov 16 2010
a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011
a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012
a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013
Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:
1  2  2  3  4  4  5  6  6...
      1  2  2  3  4  4  5...
            1  2  2  3  4...
                  1  2  2...
                        1...
------------------------------
1  2  3  5  7  9 12 15 18... - L. Edson Jeffery, Jul 30 2014
a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020
Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + 18*x^9 + ...
1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).
[n] a(n)
--------
[1] 1
[2] 2
[3] 3
[4] 1 + 4
[5] 2 + 5
[6] 3 + 6
[7] 1 + 4 + 7
[8] 2 + 5 + 8
[9] 3 + 6 + 9
a(7) = floor(2/3) +floor(3/3) +floor(4/3) +floor(5/3) +floor(6/3) +floor(7/3) +floor(8/3) +floor(9/3) = 12. - _Bruno Berselli_, Aug 29 2013

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, MR2224983(2007b:90134), 2007.
Hansraj Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
Richard K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
Peter M. Chema, Illustration of first 27 terms as corners of a double hexagon spiral from 0
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 207.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, Discrete Optimization 3, pp. 165-173, 2006.
Michael Somos, Somos Polynomials.
Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), Article 98.1.4.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 7.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
Index entries for sequences related to Lyndon words.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000031, A001037, A008748, A051168.
Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
Cf. A058937.
A column of triangle A011847.
Cf. A000217, A130205.
Cf. A258708.
A001399 counts 3-part partitions, ranked by A014612.
A337483 counts either weakly increasing or weakly decreasing triples.
A337484 counts neither strictly increasing nor strictly decreasing triples.
A014311 ranks 3-part compositions, with strict case A337453.
Cf. A007052, A007304, A011782, A069905, A156040, A218004.

a001840 n = a001840_list !! n
a001840_list = scanl (+) 0 a008620_list
-- Reinhard Zumkeller, Apr 16 2012

[ n le 2 select n else n*(n+1)/2-Self(n-1)-Self(n-2): n in [1..58] ];  // Klaus Brockhaus, Oct 01 2009

A001840 := n->floor((n+1)*(n+2)/6);
A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
seq(floor(binomial(n-1,2)/3), n=3..61); # Zerinvary Lajos, Jan 12 2009
A001840 :=  n -> add(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A001840(n), n = 0 .. 58); # Peter Luschny, Jul 06 2011

a[0]=0; a[1]=1; a[n_]:= a[n]= n(n+1)/2 -a[n-1] -a[n-2]; Table[a[n], {n,0,100}]
f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)
CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* Robert G. Wilson v *)
a[ n_] := With[{m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ x /((1 - x^3) (1 - x)^2), {x, 0, m}]]; (* Michael Somos, Jul 11 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},60] (* Harvey P. Dale, Jul 25 2011 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = (n+1) * (n+2) \ 6}; /* Michael Somos, Feb 11 2004 */

[binomial(n, 2) // 3 for n in range(2, 61)] # Zerinvary Lajos, Dec 01 2009

a(n) = (A000217(n+1) - A022003(n-1))/3;
a(n) = (A016754(n+1) - A010881(A016754(n+1)))/24;
a(n) = (A033996(n+1) - A010881(A033996(n+1)))/24.
Euler transform of length 3 sequence [2, 0, 1].
a(3*k-1) = k*(3*k + 1)/2;
a(3*k)   = 3*k*(k + 1)/2;
a(3*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).
a(n+1) = a(n) + A008620(n) = A002264(n+3). - Reinhard Zumkeller, Aug 01 2002
From Michael Somos, Feb 11 2004: (Start)
G.f.: x / ((1-x)^2 * (1-x^3)).
a(n) = 1 + a(n-1) + a(n-3) - a(n-4).
a(-3-n) = a(n). (End)
a(n) = a(n-3) + n for n > 2; a(0)=0, a(1)=1, a(2)=2. - Paul Barry, Jul 14 2004
a(n) = binomial(n+3, 3)/(n+3) + cos(2*Pi*(n-1)/3)/9 + sqrt(3)sin(2*Pi*(n-1)/3)/9 - 1/9. - Paul Barry, Jan 01 2005
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} k*(cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3).
a(n) = Sum_{k=0..floor(n/3)} n-3*k. (End)
For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
G.f.: x/(1 + x + x^2)/(1 - x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = (4 + 3*n^2 + 9*n)/18 + ((n mod 3) - ((n-1) mod 3))/9. - Klaus Brockhaus, Oct 01 2009
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Jul 25 2011
a(n) = A214734(n + 2, 1, 3). - Renzo Benedetti, Aug 27 2012
G.f.: x*G(0), where G(k) = 1 + x*(3*k+4)/(3*k + 2 - 3*x*(k+2)*(3*k+2)/(3*(1+x)*k + 6*x + 4 - x*(3*k+4)*(3*k+5)/(x*(3*k+5) + 3*(k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2013
Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - Richard R. Forberg, Jul 24 2013
a(n) = Sum_{i=0..n} floor((i+2)/3). - Bruno Berselli, Aug 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = n/2 + floor(n^2/3 + 2/3)/2. - Bruno Berselli, Jan 23 2017
a(n) + a(n+1) = A000212(n+2). - R. J. Mathar, Jan 14 2021
Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
E.g.f.: (exp(x)*(4 + 12*x + 3*x^2) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Stefano Spezia, Apr 05 2023

A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

Original entry on oeis.org

1, 5, 10, 40, 150, 624, 2580, 11160, 48750, 217000, 976248, 4438920, 20343700, 93900240, 435959820, 2034504992, 9536718750, 44878791360, 211927516500, 1003867701480, 4768371093720, 22706531339280

Offset: 0

N. J. A. Sloane

Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. = A269843 as a product_{n>=2} zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 5 colors. - Herbert Kociemba, Nov 25 2016

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
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 = 0..1435 (terms 0..200 from T. D. Noe)
Alin Bostan, Alexander Marynych, and Kilian Raschel, On the least common multiple of several random~integers, arXiv:1901.03002 [math.PR], 2019.
Jeremie Detrey, P. J. Spaenlehauer, and P. Zimmermann, Computing the rho constant, Preprint 2016.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Dionisel Y. Regalado and Rodel Azura, An Analytic Approximation to the Density of Twin Primes, Recoletos Multidisciplinary Research Journal (2019) Vol. 6, No. 2.
G. J. Simmons, The number of irreducible polynomials of degree n over GF(p), Amer. Math. Monthly, 77 (1970), 743-745.
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

Cf. A001037, A054720, A002105.
5th column of A074650. - Alois P. Heinz, Aug 08 2008
Cf. A008683, A000351, A027750.

a001692 n = flip div n $ sum $
            zipWith (*) (map a008683 divs) (map a000351 $ reverse divs)
            where divs = a027750_row n
-- Reinhard Zumkeller, Oct 07 2015

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*5^(n/d)/n, {d, Divisors[n]}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Mar 11 2014 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,5],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)*5^(n/d))/n,1) \\ Charles R Greathouse IV, Jun 15 2011

a(n) = Sum_{d|n} mu(d)*5^(n/d)/n, for n>0.

G.f.: k=5, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

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

cp's OEIS Frontend

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.

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A059966 a(n) = (1/n) * Sum_{ d divides n } mu(n/d) * (2^d - 1).

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A006918 a(n) = binomial(n+3, 3)/4 for odd n, n*(n+2)*(n+4)/24 for even n.

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A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

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A027375 Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n.

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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.

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A006918 a(n) = binomial(n+3, 3)/4 for odd n, n(n+2)(n+4)/24 for even n.