A192330 Minimum number of endpoints of a tree so that there exists a zero-entropy map defined on it having a period n orbit.

1, 2, 3, 2, 5, 3, 7, 2, 6, 5, 11, 3, 13, 7, 10, 2, 17, 6, 19, 5, 14, 11, 23, 3, 20, 13, 15, 7, 29, 10, 31, 2, 22, 17, 28, 6, 37, 19, 26, 5, 41, 14, 43, 11, 25, 23, 47, 3, 42, 20, 34, 13, 53, 15, 44, 7, 38, 29, 59, 10, 61, 31, 35, 2, 52, 22, 67, 17, 46, 28, 71, 6, 73, 37

Offset

1,2

Comments

The topological entropy of a continuous map from a compact metric space into itself is a quantitative measure of the complexity of the dynamical system defined by the iteration of the map. See Adler, Konheim, McAndrew reference.

Links

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.

E. Barrabés, D. Juher, The minimum tree for a given zero entropy period, Int. J. Math. Math. Sci. 2005:19 (2005), pp. 3025-3033.

Formula

a(n) = n - Sum_{i=2..k} Product_{j=i..k} s_j, where n = s_1*s_2*...*s_k with s_i primes and s_i <= s_{i+1}.

Example

a(2^n)=2 for n > 0, a(p)=p for p prime, a(k*2^j) = a(k) for k > 0, j >= 0.

Prog

(Magma) A192330:=func< n | n-s where s:=w eq [] select 0 else &+w where w:=[ &*[ v[i]: i in [k..#v] ]: k in [2..#v] ] where v:=&cat[ [ f[j, 1]: i in [1..f[j, 2]] ]: j in [1..#f] ] where f:=Factorization(n) >; [ A192330(n): n in [1..75] ]; // Klaus Brockhaus, Jul 02 2011
(PARI) A192330(n)=
{local(f=factor(n), v=[], k, s); for(j=1, #f[, 2], for(i=1, f[j, 2], v=concat(v, f[j, 1]))); k=#v; s=sum(i=2, k, prod(j=i, k, v[j])); n-s}
vector(75, n, A192330(n)) \\ Klaus Brockhaus, Jul 02 2011

Crossrefs

Cf. A006948 (zero-entropy permutations of length n), A109395 (denominator of phi(n)/n, phi(n)=A000010(n) is the Euler totient function).

Sequence in context: A303077 A347619 A164858 * A320028 A327076 A215041

Adjacent sequences: A192327 A192328 A192329 * A192331 A192332 A192333

Keyword

nonn

Author

David Juher, Jun 28 2011

Status

approved