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A192330
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Minimum number of endpoints of a tree so that there exists a zero-entropy map defined on it having a period n orbit.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.
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FORMULA
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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}.
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EXAMPLE
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a(2^n)=2 for n > 0, a(p)=p for p prime, a(k*2^j) = a(k) for k > 0, j >= 0.
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PROG
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(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
{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}
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CROSSREFS
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Cf. A006948 (zero-entropy permutations of length n), A109395 (denominator of phi(n)/n, phi(n)=A000010(n) is the Euler totient function).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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