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 A027375 Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n. 51
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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; 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 two last 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) REFERENCES 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. 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 LINKS 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, 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. 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. FORMULA 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 EXAMPLE a(3) = 6 = |{ 001, 010, 011, 100, 101, 110 }|. - corrected by Geoffrey Critzer, Dec 07 2014 MAPLE with(numtheory): A027375 :=n->add( mobius(d)*2^(n/d), d = divisors(n)); # N. J. A. Sloane, Sep 25 2012 MATHEMATICA Table[ Apply[ Plus, MoebiusMu[ n / Divisors[n] ]*2^Divisors[n] ], {n, 1, 32} ] a=0; a[n_] := DivisorSum[n, MoebiusMu[n/#]*2^#&]; Array[a, 40, 0] (* Jean-François Alcover, Dec 01 2015 *) PROG (PARI) a(n) = sumdiv(n, d, moebius(n/d)*2^d); (Haskell) a027375 n = n * a001037 n  -- Reinhard Zumkeller, Feb 01 2013 (Python) 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 xrange(101)] # Indranil Ghosh, Jun 28 2017 CROSSREFS A038199 and A056267 are essentially the same sequence with different initial terms. Cf. A020921, A216953. Column k=2 of A143324. Sequence in context: A216215 A052994 A088219 * A059727 A103872 A216641 Adjacent sequences:  A027372 A027373 A027374 * A027376 A027377 A027378 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 14 01:36 EDT 2019. Contains 327994 sequences. (Running on oeis4.)