cp's OEIS Frontend

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

%I A192330 #20 Sep 08 2022 08:45:57
%S A192330 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,
%T A192330 29,10,31,2,22,17,28,6,37,19,26,5,41,14,43,11,25,23,47,3,42,20,34,13,
%U A192330 53,15,44,7,38,29,59,10,61,31,35,2,52,22,67,17,46,28,71,6,73,37
%N A192330 Minimum number of endpoints of a tree so that there exists a zero-entropy map defined on it having a period n orbit.
%C A192330 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.
%H A192330 Klaus Brockhaus, <a href="/A192330/b192330.txt">Table of n, a(n) for n = 1..10000</a>
%H A192330 R. L. Adler, A. G. Konheim and M. H. McAndrew, <a href="https://doi.org/10.1090/S0002-9947-1965-0175106-9">Topological entropy</a>, Trans. Amer. Math. Soc. 114 (1965), 309-319.
%H A192330 E. Barrabés, D. Juher, <a href="http://dx.doi.org/10.1155/IJMMS.2005.3025">The minimum tree for a given zero entropy period</a>, Int. J. Math. Math. Sci. 2005:19 (2005), pp. 3025-3033.
%F A192330 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}.
%e A192330 a(2^n)=2 for n > 0, a(p)=p for p prime, a(k*2^j) = a(k) for k > 0, j >= 0.
%o A192330 (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
%o A192330 (PARI) A192330(n)=
%o A192330 {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}
%o A192330 vector(75, n, A192330(n)) \\ _Klaus Brockhaus_, Jul 02 2011
%Y A192330 Cf. A006948 (zero-entropy permutations of length n), A109395 (denominator of phi(n)/n, phi(n)=A000010(n) is the Euler totient function).
%K A192330 nonn
%O A192330 1,2
%A A192330 _David Juher_, Jun 28 2011