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A001037
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Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.
(Formerly M0116 N0046 N0287)
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227
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1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008, 9586395, 18512790, 35790267, 69273666, 134215680, 260300986, 505286415, 981706806, 1908866960, 3714566310, 7233615333, 14096302710, 27487764474
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OFFSET
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0,2
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COMMENTS
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Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence.
This sequence also represents the number N of cycles of length L in a digraph under x^2 seen modulo a Mersenne prime M_q=2^q-1. This number does not depend on q and L is any divisor of q-1. See Theorem 5 and Corollary 3 of the Shallit and Vasiga paper: N=sum(eulerphi(d)/order(d,2)) where d is a divisor of 2^(q-1)-1 such that order(d,2)=L. - Tony Reix, Nov 17 2005
Except for a(0) = 1, Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 7th (rightmost) column. Other columns of the table include (but are not identified as) A006206-A006208. - Jonathan Vos Post, Jun 18 2007
"Number of binary Lyndon words" means: number of binary strings inequivalent modulo rotation (cyclic permutation) of the digits and not having a period smaller than n. This provides a link to A103314, since these strings correspond to the inequivalent zero-sum subsets of U_m (m-th roots of unity) obtained by taking the union of U_n (n|m) with 0 or more U_d (n | d, d | m) multiplied by some power of exp(i 2Pi/n) to make them mutually disjoint. (But not all zero-sum subsets of U_m are of that form.) - M. F. Hasler, Jan 14 2007
Also the number of dynamical cycles of period n of a threshold Boolean automata network which is a quasi-minimal positive circuit of size a multiple of n and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Feb 25 2009
Also, the number of periodic points with (minimal) period n in the iteration of the tent map f(x):=2min{x,1-x} on the unit interval. - Pietro Majer, Sep 22 2009
Number of distinct cycles of minimal period n in a shift dynamical system associated with a totally disconnected hyperbolic iterated function system (see Barnsley link). - Michel Marcus, Oct 06 2013
For n > 0, a(n) is also the number of orbits of size n of the transform associated to the Kolakoski sequence A000002 (and this is true for any map with 2^n periodic points of period n). The Kolakoski 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 periodic points of the orbit of size 2. A027375(n) = n*a(n) gives the number of periodic points of minimal period n.
For n > 1, this sequence is equal to A059966 and to A060477, and for n = 1, a(1) = A059966(1)+1 = A060477(1)-1; this because the n-th term of all 3 sequences is equal to (1/n)*sum_{d|n} mu(n/d)*(2^d+e), with e = -1/0/1 for resp. A059966/this sequence/A060477, and sum_{d|n} mu(n/d) equals 1 for n = 1 and 0 for all n > 1. (End)
Warning: A000031 and A001037 are easily confused, since they have similar formulas.
Following Kam Cheong Au (2020), let d(w,N) be the dimension of the Q-span of weight w and level N of colored multiple zeta values (CMZV). Here Q are the rational numbers.
Deligne's bound says that d(w,N) <= D(w,N), where 1 + Sum_{w >= 1} D(w,N)*t^w = (1 - a*t + b*t^2)^(-1) when N >= 3, where a = phi(N)/2 + omega(N) and b = omega(N) - 1 (with omega(N) = A001221(N) being the number of distinct primes of N).
For N = 3, a = phi(3)/2 + omega(3) = 2/2 + 1 = 2 and b = omega(3) - 1 = 0. It follows that D(w, N=3) = A000079(w) = 2^w.
For some reason, Kam Cheong Au (2020) assumes Deligne's bound is tight, i.e., d(w,N) = D(w,N). He sets Sum_{w >= 1} c(w,N)*t^w = log(1 + Sum_{w >= 1} d(w,N)*t^w) = log(1 + Sum_{w >= 1} D(w,N)*t^w) = -log(1 - a*t + b*t^2) for N >= 3.
For N = 3, we get that c(w, N=3) = A000079(w)/w = 2^w/w.
He defines d*(w,N) = Sum_{k | w} (mu(k)/k)*c(w/k,N) to be the "number of primitive constants of weight w and level N". (Using the terminology of A113788, we may perhaps call d*(w,N) the number of irreducible colored multiple zeta values at weight w and level N.)
Using standard techniques of the theory of g.f.'s, we can prove that Sum_{w >= 1} d*(w,N)*t^w = Sum_{s >= 1} (mu(s)/s) Sum_{k >= 1} c(k,N)*(t^s)^k = -Sum_{s >= 1} (mu(s)/s)*log(1 - a*t^s + b*t^(2*s)).
For N = 3, we saw that a = 2 and b = 0, and hence d*(w, N=3) = a(w) = Sum_{k | w} (mu(k)/k) * 2^(w/k) / (w/k) = (1/w) * Sum_{k | w} mu(k) * 2^(w/k) for w >= 1. See Table 1 on p. 6 in Kam Cheong Au (2020). (End)
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REFERENCES
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Michael F. Barnsley, Fractals Everywhere, Academic Press, San Diego, 1988, page 171, Lemma 3.
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie. On the digraph defined by squaring mod m, when m has primitive roots. Congr. Numer. 82 (1991), 167-177.
P. J. Freyd and A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. See 1.925.
M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983, pp. 65, 79.
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
Guy Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
M. R. Nester, (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in entries N0046 and N0287).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Émilie Charlier, Manon Philibert, and Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. Also J. Comb. Thy. A, 167 (2019), 60-90.
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FORMULA
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For n >= 1:
a(n) = (1/n)*Sum_{d | n} mu(n/d)*2^d.
2^n = Sum_{d | n} d*a(d).
G.f.: 1 - Sum_{n >= 1} moebius(n)*log(1 - 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
For n >= 1:
a(n) = (1/n)*Sum_{k=1..n} mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
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EXAMPLE
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Binary strings (Lyndon words, cf. A102659):
a(0) = 1 = #{ "" },
a(1) = 2 = #{ "0", "1" },
a(2) = 1 = #{ "01" },
a(3) = 2 = #{ "001", "011" },
a(4) = 3 = #{ "0001", "0011", "0111" },
a(5) = 6 = #{ "00001", "00011", "00101", "00111", "01011", "01111" }.
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MAPLE
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with(numtheory): A001037 := proc(n) local a, d; if n = 0 then RETURN(1); else a := 0: for d in divisors(n) do a := a+mobius(n/d)*2^d; od: RETURN(a/n); fi; end;
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MATHEMATICA
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f[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[n/d]*2^d/n)]; Array[f, 32]
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PROG
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(PARI) A001037(n)=if(n>1, sumdiv(n, d, moebius(d)*2^(n/d))/n, n+1) \\ Edited by M. F. Hasler, Jan 11 2016
(PARI) {a(n)=polcoeff(1-sum(k=1, n, moebius(k)/k*log(1-2*x^k+x*O(x^n))), n)} \\ Paul D. Hanna, Oct 13 2010
(PARI) a(n)=if(n>1, my(s); forstep(i=2^n+1, 2^(n+1), 2, s+=polisirreducible(Mod(1, 2) * Pol(binary(i)))); s, n+1) \\ Charles R Greathouse IV, Jan 26 2012
(Haskell)
a001037 0 = 1
a001037 n = (sum $ map (\d -> (a000079 d) * a008683 (n `div` d)) $
a027750_row n) `div` n
(Python)
from sympy import divisors, mobius
def a(n): return sum(mobius(d) * 2**(n//d) for d in divisors(n))/n if n>1 else n + 1 # Indranil Ghosh, Apr 26 2017
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CROSSREFS
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Row sums of A051168, which gives the number of Lyndon words with fixed number of zeros and ones.
Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.
See also A102659 for the list of binary Lyndon words themselves.
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KEYWORD
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nonn,core,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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