A060220 Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

17, 136, 1632, 20808, 283968, 4022064, 58619808, 871959240, 13176430176, 201599248032, 3115626937056, 48551851084080, 761890617915840, 12026987582075856, 190828203433892736, 3041324491793194440, 48661191875666868480, 781282469552728498992, 12582759772902701307744

Offset: 1

Author

Thomas Ward, Mar 21 2001

Keywords

• easy
• nonn

Comments

Number of monic irreducible polynomials of degree n over GF(17). - Andrew Howroyd, Dec 10 2017

Examples

			a(2)=136 since there are 289 points of period 2 in the full 17-shift and 17 fixed points, so there must be (289-17)/2 = 136 orbits of length 2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..810
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
• T. Ward, Exactly realizable sequences

Crossrefs

Column 17 of A074650.

Cf. A001026, A008683.

Programs

Magma

A060220:= func< n | (1/n)*(&+[MoebiusMu(d)*(17)^Floor(n/d): d in Divisors(n)]) >;
[A060220(n): n in [1..40]]; // G. C. Greubel, Sep 13 2024

Mathematica

A060220[n_]:= DivisorSum[n, (17)^(n/#)*MoebiusMu[#] &]/n;
Table[A060220[n], {n,40}] (* G. C. Greubel, Sep 13 2024 *)

PARI

a001024(n) = 17^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017

SageMath

def A060220(n): return (1/n)*sum(moebius(k)*(17)^(n/k) for k in (1..n) if (k).divides(n))
[A060220(n) for n in range(1,41)] # G. C. Greubel, Sep 13 2024

Formula

a(n) = (1/n)* Sum_{d|n} mu(d)*A001026(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 17*x^k))/k. - Ilya Gutkovskiy, May 20 2019

Extensions

More terms from Michel Marcus, Sep 11 2017