A027375 Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n.

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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; 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)

Examples

			a(3) = 6 = |{ 001, 010, 011, 100, 101, 110 }|. - corrected by _Geoffrey Critzer_, Dec 07 2014
		

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.

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 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.

Crossrefs

A038199 and A056267 are essentially the same sequence with different initial terms.

Cf. A020921, A216953.

Column k=2 of A143324.

Programs

Haskell

a027375 n = n * a001037 n  -- Reinhard Zumkeller, Feb 01 2013

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]=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);

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 range(101)]) # Indranil Ghosh, Jun 28 2017

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

G.f.: 2 * Sum_{k>=1} mu(k)*x^k/(1 - 2*x^k). - Ilya Gutkovskiy, Nov 11 2019