|
| |
|
|
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)
|
|
19
|
|
|
|
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. 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). - D. E. 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, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363). [From Emeric Deutsch, Aug 13 2008]
Moebius transform (A054525) of [1, 2, 4, 8,...]; = row sums of triangle A143424. [From Gary W. Adamson, Aug 14 2008]
|
|
|
REFERENCES
|
E. Deutsch and Lafayette College Problem Group, Problem 11161, 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 (1964), 241-260.
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
|
T. D. Noe, Table of n, a(n) for n=1..300
R. Munafo, Enumeration of Period-N Mu-Atoms
Index entries for sequences related to Lyndon words
|
|
|
FORMULA
|
a(n) = sum_{d|n} mu(n/d)*2^(d-1). 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). - Emeric Deutsch, Apr 27 2007
|
|
|
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>.
|
|
|
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}] (* From Jean-François Alcover, Feb 03 2012, after Pari *)
|
|
|
PROG
|
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*2^(d-1))
|
|
|
CROSSREFS
|
Cf. A003239, A022553, A034738, A035928, A000837, A216954.
Cf. A054525, A143424 [From Gary W. Adamson, Aug 14 2008]
Cf. A008683, A178472, A167606, A056278.
Equals A027375/2.
Sequence in context: A134774 A165729 A056278 * A161625 A069712 A076971
Adjacent sequences: A000737 A000738 A000739 * A000741 A000742 A000743
|
|
|
KEYWORD
|
nonn,nice,easy,changed
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
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
|
| |
|
|
