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A357005
Smallest k that is cyclically equivalent (see Comment for definition) to n.
4
1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 11, 9, 11, 11, 15, 16, 17, 17, 19, 17, 19, 19, 23, 17, 19, 19, 23, 19, 23, 23, 31, 32, 33, 34, 35, 36, 37, 37, 39, 34, 37, 42, 43, 37, 45, 43, 47, 33, 35, 37, 39, 37, 43, 45, 47, 35, 39, 43, 47, 39, 47, 47, 63, 64, 65, 65, 67, 65
OFFSET
1,2
COMMENTS
Two positive integers k and n are cyclically equivalent (as defined by Hladnik, Marušič, and Pisanski, 2002) if they have the same number m of binary digits, and there exist integers s and t such that gcd(s,m) = 1 and the map x -> s*x+t mod m maps the set of exponents of 2 occurring in the binary expansion of n bijectively to the corresponding set for k. (In particular, A000120(k) = A000120(n).) For example, k = 17 = 2^0 + 2^4 and n = 18 = 2^1 + 2^4 are cyclically equivalent, because the map x -> 2*x+2 mod 5 maps {1,4} to {0,4}.
The fixed points are the terms of A357006.
The number of fixed points n in the interval 2^(m-1) <= n < 2^m equals A002729(m)-1.
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Milan Hladnik, Dragan Marušič, and Tomaž Pisanski, Cyclic Haar graphs, Discrete Mathematics 244 (2002), 137-152.
FORMULA
a(a(n)) = a(n).
a(n) = A357004(n) for n <= 146, but a(147) = 147 > 141 = A357004(147).
A357004(n) <= a(n) <= A163382(n).
PROG
(Python)
from math import gcd
def A357005(n):
p=[int(d) for d in format(n, 'b')]
m=len(p)
p0=min([p[(k*i+j)%m] for i in range(m)] for k in range(1, m+1) if gcd(k, m)==1 for j in range(m) if p[j])
return sum(p0[i]*2**(m-1-i) for i in range(m))
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved