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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Also number of compositions of n into relatively prime parts (that is, the gcd of all the parts is 1). Also number of subsets of {1,2,..,n} containing n and consisting of relatively prime numbers. - Vladeta Jovovic, Aug 13 2003

Also number of perfect parity patterns that have exactly n columns (see A118141). - Don Knuth, May 11 2006

a(n) is odd if and only if n is squarefree (Tim Keller). - Emeric Deutsch, Apr 27 2007

a(n) is a multiple of 3 for all n>=3 (see Problem 11161 link). - Emeric Deutsch, Aug 13 2008

Row sums of triangle A143424. - Gary W. Adamson, Aug 14 2008

a(n) is the number of monic irreducible polynomials with nonzero constant coefficient in GF(2)[x] of degree n. - Michel Marcus, Oct 30 2016

a(n) is the number of aperiodic compositions of n, the number of compositions of n with relatively prime parts, and the number of compositions of n with relatively prime run-lengths. - Gus Wiseman, Dec 21 2017

Examples

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

References

• H. O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag; contribution by A. Douady, p. 165.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Crossrefs

Cf. A000837, A003239, A008683, A008965, A022553, A034738, A035928, A038199, A051168, A054525, A056267, A059966, A143424, A167606, A178472, A216954, A228369, A294859, A296302.

Equals A027375/2.

See A056278 for a variant.

First differences of A085945.

Column k=2 of A143325.

Row sums of A101391.

Programs

Maple

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

Mathematica

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

PARI

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

Python

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

Formula

a(n) = Sum_{d|n} mu(n/d)*2^(d-1), Mobius transform of A011782. Furthermore, Sum_{d|n} a(d) = 2^(n-1).

a(n) = A027375(n)/2 = A038199(n)/2.

a(n) = Sum_{k=0..n} A051168(n,k)*k. - Max Alekseyev, Apr 09 2013

Recurrence relation: a(n) = 2^(n-1) - Sum_{d|n,d>1} a(n/d). (Lafayette College Problem Group; see the Maple program and Iglesias eq (6)). - Emeric Deutsch, Apr 27 2007

G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 2*x^k). - Ilya Gutkovskiy, Oct 24 2018

G.f. satisfies Sum_{n>=1} A( (x/(1 + 2*x))^n ) = x. - Paul D. Hanna, Apr 02 2025

Extensions

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

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