A000740 -id:A000740 - cp's OEIS frontend

A003242 Number of compositions of n such that no two adjacent parts are equal (these are sometimes called Carlitz compositions).

1, 1, 1, 3, 4, 7, 14, 23, 39, 71, 124, 214, 378, 661, 1152, 2024, 3542, 6189, 10843, 18978, 33202, 58130, 101742, 178045, 311648, 545470, 954658, 1670919, 2924536, 5118559, 8958772, 15680073, 27443763, 48033284, 84069952, 147142465, 257534928, 450748483, 788918212

Offset: 0

E. Rodney Canfield

			From _Joerg Arndt_, Oct 27 2012:  (Start)
The 23 such compositions of n=7 are
[ 1]  1 2 1 2 1
[ 2]  1 2 1 3
[ 3]  1 2 3 1
[ 4]  1 2 4
[ 5]  1 3 1 2
[ 6]  1 3 2 1
[ 7]  1 4 2
[ 8]  1 5 1
[ 9]  1 6
[10]  2 1 3 1
[11]  2 1 4
[12]  2 3 2
[13]  2 4 1
[14]  2 5
[15]  3 1 2 1
[16]  3 1 3
[17]  3 4
[18]  4 1 2
[19]  4 2 1
[20]  4 3
[21]  5 2
[22]  6 1
[23]  7
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 191.

Alois P. Heinz, Table of n, a(n) for n = 0..4100 (first 501 terms from Christian G. Bower)
L. Carlitz, Restricted Compositions, Fibonacci Quarterly, 14 (1976) 254-264.
Sylvie Corteel, Paweł Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 42 and 117.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 201
F. Harary & R. W. Robinson, The number of achiral trees, Jnl. Reine Angewandte Mathematik 278 (1975), 322-335. (Annotated scanned copy)
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Chapter 8.

Cf. A106351, A114900, A114902.
Cf. A096568, A011782, A106356. - Franklin T. Adams-Watters, May 27 2010
Row sums of A232396, A241701.
Cf. A241902.
Column k=1 of A261960.
Cf. A048272.
Compositions with adjacent parts coprime are A167606.
The complement is counted by A261983.
Cf. A000740, A005251, A032020, A114901, A178470, A261041, A274174, A329738, A329863.

a003242 n = a003242_list !! n
a003242_list = 1 : f [1] where
   f xs = y : f (y : xs) where
          y = sum $ zipWith (*) xs a048272_list
-- Reinhard Zumkeller, Nov 04 2015

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(`if`(j=i, 0, b(n-j, `if`(j<=n-j, j, 0))), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 27 2014

A048272[n_] := Total[If[OddQ[#], 1, -1]& /@ Divisors[n]]; a[n_] := a[n] = Sum[A048272[k]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 38}](* Jean-François Alcover, Nov 25 2011, after Vladeta Jovovic *)
nmax = 50; CoefficientList[Series[1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 07 2020 *)
Table[Count[Flatten[Permutations/@IntegerPartitions[n],1],?(FreeQ[Differences[#],0]&)],{n,0,20}] (* The program generates the first 21 terms of the sequence. *) (* _Harvey P. Dale, Nov 23 2024 *)

N = 66;  x = 'x + O('x^N);  p=2;
gf = 1/(1-sum(k=1,N, x^k/(1-x^k)-p*x^(k*p)/(1-x^(k*p))));
Vec(gf)  /* Joerg Arndt, Apr 28 2013 */

a(n) = Sum_{k=1..n} A048272(k)*a(n-k), n>1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/(1 - Sum_{k>0} x^k/(1+x^k)).
a(n) ~ c r^n where c is approximately 0.456387 and r is approximately 1.750243. (Formula from Knopfmacher and Prodinger reference.) - Franklin T. Adams-Watters, May 27 2010. With better precision: r = 1.7502412917183090312497386246398158787782058181381590561316586... (see A241902), c = 0.4563634740588133495321001859298593318027266156100046548066205... - Vaclav Kotesovec, Apr 30 2014
G.f. is the special case p=2 of 1/(1 - Sum_{k>0} (z^k/(1-z^k) - p*z^(k*p)/(1-z^(k*p)))), see A129922. - Joerg Arndt, Apr 28 2013
G.f.: 1/(1 - x * (d/dx) log(Product_{k>=1} (1 + x^k)^(1/k))). - Ilya Gutkovskiy, Oct 18 2018
Moebius transform of A329738. - Gus Wiseman, Nov 27 2019
For n>=2, a(n) = A128695(n) - A091616(n). - Vaclav Kotesovec, Jul 07 2020

More terms from David W. Wilson

A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 100, 119, 167, 209, 296, 347, 489, 582, 775, 945, 1254, 1481, 1951, 2334, 2980, 3580, 4564, 5386, 6841, 8118, 10085, 12012, 14862, 17526, 21636, 25524, 31082, 36694, 44582, 52255, 63260, 74170, 88931, 104302

Offset: 0

N. J. A. Sloane

Starting (1, 1, 2, 3, 6, 7, 14, ...), = row sums of triangle A137585. - Gary W. Adamson, Jan 27 2008
Triangle A168532 has aerated variants of this sequence in each column starting with offset 1, row sums = A000041. -  Gary W. Adamson, Nov 28 2009
A partition is aperiodic iff its multiplicities are relatively prime, i.e., its Heinz number (A215366) is not a perfect power (A007916). - Gus Wiseman, Dec 19 2017
This sequence is monotonically increasing; each partition of n-1 can have a part of size 1 added to it to get a partition counted in a(n). - Franklin T. Adams-Watters, Jul 24 2020

			Of the 11 partitions of 6, we must exclude 6, 4+2, 3+3 and 2+2+2, so a(6) = 11 - 4 = 7.
For n=6, 2+2+1+1 is periodic because it can be written 2*(2+1), similarly 1+1+1+1+1+1, 3+3 and 2+2+2.
The a(6) = 7 partitions into relatively prime parts are (51), (411), (321), (3111), (2211), (21111), (111111). The a(6) = 7 aperiodic partitions are (6), (51), (42), (411), (321), (3111), (21111). - _Gus Wiseman_, Dec 19 2017

H. W. Gould, personal communication.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Mohamed El Bachraoui, On the Parity of p(n,3) and p_psi(n,3), Contributions to Discrete Mathematics, Vol. 5.2 (2010).
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Mircea Merca and Maxie D. Schmidt, Generating Special Arithmetic Functions by Lambert Series Factorizations, arXiv:1706.00393 [math.NT], 2017. See Remark 3.4.
N. J. A. Sloane, Transforms

Cf. A000041, A018783.

Cf. A000740, A007916, A047968, A055892, A100953, A137585, A168532, A281116.

p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]]; g[n_, j_] := Apply[GCD, Part[p[n], j]]; h[n_] := Table[g[n, j], {j, 1, l[n]}]; Join[{1}, Table[Count[h[n], 1], {n, 1, 20}]]
(* Clark Kimberling, Mar 09 2012 *)
a[0] = 1; a[n_] := Sum[ MoebiusMu[n/d] * PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 03 2013 *)

N=66; x='x+O('x^N); gf=2+sum(n=1,N, (1/eta(x^n))*moebius(n)); Vec(gf) \\ Joerg Arndt, May 11 2013

print1("1, "); for(n=1,46,my(s=0);forpart(X=n,s+=gcd(X)==1);print1(s,", ")) \\ Hugo Pfoertner, Mar 27 2020

from sympy import npartitions, mobius, divisors
def a(n): return 1 if n==0 else sum(mobius(n//d)*npartitions(d) for d in divisors(n)) # Indranil Ghosh, Apr 26 2017

Möbius transform of A000041. - Christian G. Bower, Jun 11 2000
Product_{n>0} 1/(1-q^n) = 1 + Sum_{n>0} a(n)*q^n/(1-q^n). - Mamuka Jibladze, Nov 14 2015
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019
a(n) <= p(n) <= a(n+1), where p(n) is the number of partitions of n (A000041). - Franklin T. Adams-Watters, Jul 24 2020

Corrected and extended by David W. Wilson, Aug 15 1996

Additional name from Christian G. Bower, Jun 11 2000

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

A329739 Number of compositions of n whose run-lengths are all different.

Original entry on oeis.org

1, 1, 2, 2, 5, 8, 10, 20, 28, 41, 62, 102, 124, 208, 278, 426, 571, 872, 1158, 1718, 2306, 3304, 4402, 6286, 8446, 11725, 15644, 21642, 28636, 38956, 52296, 70106, 93224, 124758, 165266, 218916, 290583, 381706, 503174, 659160, 865020, 1124458, 1473912, 1907298

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(7) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (211)   (221)    (222)     (223)
                    (1111)  (311)    (411)     (322)
                            (1112)   (1113)    (331)
                            (2111)   (3111)    (511)
                            (11111)  (11112)   (1114)
                                     (21111)   (1222)
                                     (111111)  (2221)
                                               (4111)
                                               (11113)
                                               (11122)
                                               (22111)
                                               (31111)
                                               (111112)
                                               (111211)
                                               (112111)
                                               (211111)
                                               (1111111)

The normal case is A329740.
The case of partitions is A098859.
Strict compositions are A032020.
Compositions with relatively prime run-lengths are A000740.
Compositions with distinct multiplicities are A242882.
Compositions with distinct differences are A325545.
Compositions with equal run-lengths are A329738.
Compositions with normal run-lengths are A329766.
Cf. A003242, A008965, A059966, A098504, A098859, A107429, A175342, A238130, A328592, A329745.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Length/@Split[#]&]],{n,0,10}]

a(21)-a(26) from Giovanni Resta, Nov 22 2019

a(27)-a(43) from Alois P. Heinz, Jul 06 2020

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

A078374 Number of partitions of n into distinct and relatively prime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 10, 17, 17, 23, 26, 37, 36, 53, 53, 70, 77, 103, 103, 139, 147, 184, 199, 255, 260, 339, 358, 435, 474, 578, 611, 759, 810, 963, 1045, 1259, 1331, 1609, 1726, 2015, 2200, 2589, 2762, 3259, 3509, 4058, 4416, 5119, 5488, 6364, 6882

Offset: 1

Vladeta Jovovic, Dec 24 2002

nonn

The Heinz numbers of these partitions are given by A302796, which is the intersection of A005117 (strict) and A289509 (relatively prime). - Gus Wiseman, Oct 18 2020

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(13) = 17 partitions (empty column indicated by dot, A = 10, B = 11, C = 12):
  1   .  21   31   32   51    43    53    54    73     65     75     76
                   41   321   52    71    72    91     74     B1     85
                              61    431   81    532    83     543    94
                              421   521   432   541    92     651    A3
                                          531   631    A1     732    B2
                                          621   721    542    741    C1
                                                4321   632    831    643
                                                       641    921    652
                                                       731    5421   742
                                                       821    6321   751
                                                       5321          832
                                                                     841
                                                                     931
                                                                     A21
                                                                     5431
                                                                     6421
                                                                     7321
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A047966.
A000837 is the not necessarily strict version.
A302796 gives the Heinz numbers of these partitions.
A305713 is the pairwise coprime instead of relatively prime version.
A332004 is the ordered version.
A337452 is the case without 1's.
A000009 counts strict partitions.
A000740 counts relatively prime compositions.
Cf. A007359, A101268, A289508, A289509, A291166, A298748, A337451, A337485, A337451, A337561, A337563.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]],{n,15}] (* Gus Wiseman, Oct 18 2020 *)

Moebius transform of A000009.

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n). - Ilya Gutkovskiy, Apr 26 2017

A329738 Number of compositions of n whose run-lengths are all equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 19, 24, 45, 75, 133, 215, 401, 662, 1177, 2035, 3587, 6190, 10933, 18979, 33339, 58157, 101958, 178046, 312088, 545478, 955321, 1670994, 2925717, 5118560, 8960946, 15680074, 27447350, 48033502, 84076143, 147142496, 257546243, 450748484, 788937192

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (131)    (51)
                            (212)    (123)
                            (11111)  (132)
                                     (141)
                                     (213)
                                     (222)
                                     (231)
                                     (312)
                                     (321)
                                     (1122)
                                     (1212)
                                     (2121)
                                     (2211)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Compositions with relatively prime run-lengths are A000740.
Compositions with equal multiplicities are A098504.
Compositions with equal differences are A175342.
Compositions with distinct run-lengths are A329739.
Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329745, A329766.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@Split[#]&]],{n,0,10}]

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); concat([1], vector(n, k, sumdiv(k, d, b[d])))} \\ Andrew Howroyd, Dec 30 2020

a(n) = Sum_{d|n} A003242(d).

a(n) = A329745(n) + A000005(n).

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.

A100953 Number of partitions of n into relatively prime parts such that multiplicities of parts are also relatively prime.

Original entry on oeis.org

1, 1, 0, 1, 2, 5, 5, 13, 14, 25, 28, 54, 54, 99, 105, 160, 192, 295, 315, 488, 546, 760, 890, 1253, 1404, 1945, 2234, 2953, 3459, 4563, 5186, 6840, 7909, 10029, 11716, 14843, 17123, 21635, 25035, 30981, 36098, 44581, 51370, 63259, 73223, 88739, 103048, 124752

Offset: 0

Vladeta Jovovic, Jan 11 2005

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000740, A000837, A007916, A281116, A296302.

read transforms : a000837 := [] : b000837 := fopen("b000837.txt",READ) : bfil := readline(b000837) : while StringTools[WordCount](bfil) > 0 do b := sscanf( bfil,"%d %d") ; a000837 := [op(a000837),op(2,b)] ; bfil := readline(b000837) ; od: fclose(b000837) ; a000837 := subsop(1=NULL,a000837) : a := MOBIUS(a000837) : for n from 1 to 120 do printf("%d, ",op(n,a)) ; od: # R. J. Mathar, Mar 12 2008
# second Maple program:
with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(
       mobius(n/d)*numbpart(d), d=divisors(n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
       mobius(n/d)*b(d), d=divisors(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 19 2017

Table[Length[Select[IntegerPartitions[n],And[GCD@@#===1,GCD@@Length/@Split[#]===1]&]],{n,20}] (* Gus Wiseman, Dec 19 2017 *)
b[n_] := b[n] = If[n==0, 1, Sum[
     MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]];
a[n_] := a[n] = If[n==0, 1, Sum[
     MoebiusMu[n/d]*b[d], {d, Divisors[n]}]];
a /@ Range[0, 60] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

Moebius transform of A000837.

More terms from David Wasserman and R. J. Mathar, Mar 04 2008

a(0)=1 prepended by Alois P. Heinz, Dec 19 2017

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

A003242 Number of compositions of n such that no two adjacent parts are equal (these are sometimes called Carlitz compositions).

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A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions.

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

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A329739 Number of compositions of n whose run-lengths are all different.

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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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A078374 Number of partitions of n into distinct and relatively prime parts.

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