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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.
(Formerly M2582 N1021)
199
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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
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.
R. Chapman and D. Knuth, Problem 11243, Perfect Parity Patterns, Am. Math. Monthly 115 (7) (2008) p 668, function c(n).
E. Deutsch and Lafayette College Problem Group, Problem 11161: Compositions without Common Factors, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
J. Shallit & N. J. A. Sloane, Correspondence 1974-1975
François Vigneron and Nicolae Mihalache, How to split a tera-polynomial, arXiv:2402.06083 [math.NA], 2024.
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
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
Equals A027375/2.
See A056278 for a variant.
First differences of A085945.
Column k=2 of A143325.
Row sums of A356027.
Sequence in context: A337665 A165729 A348378 * A056278 A161625 A234848
KEYWORD
nonn,nice,easy
AUTHOR
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
STATUS
approved

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