A357005 Smallest k that is cyclically equivalent (see Comment for definition) to n.

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

Cf. A000120, A002729, A163382, A267508, A357004, A357006.

Sequence in context: A136623 A031218 A357004 * A267508 A163382 A364019

Adjacent sequences: A357002 A357003 A357004 * A357006 A357007 A357008

Keyword

nonn

Author

Pontus von Brömssen, Sep 08 2022

Status

approved