A267508 -id:A267508 - cp's OEIS frontend

A233312 Terms of A114994 which are c-equivalent to "c-squares" (A020330).

0, 3, 10, 15, 36, 43, 43, 63, 136, 147, 170, 175, 147, 175, 175, 255, 528, 547, 586, 591, 586, 683, 683, 703, 547, 591, 683, 703, 591, 703, 703, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2347, 2367, 2186, 2347, 2730, 2735, 2347, 2735, 2735, 2815, 2115, 2191

Offset: 0

Vladimir Shevelev, Dec 07 2013

About c-equivalent see in comment in A233249.

a(n) is even iff A171791(n+1) is odd - holds for at least the first 1028 terms. The reason, put very briefly, is that: a(n) is even if and only if n is the double of a "fibbinary number". Cf. A267508. [Jörgen Backelin, Jan 15 2016 added by Jeremy Gardiner, Jan 26 2016]

			c-square of 5 in binary is (10)(1)(10)(1)~(10)(10)(1)(1) which is 43 in decimal. So a(5)=43.

Peter J. C. Moses, Table of n, a(n) for n = 0..2499
Index entries for sequences related to binary expansion of n

Cf. A020330, A114994, A171791, A233249, A267508, A268032.

More terms from Peter J. C. Moses, Dec 07 2013

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

Original entry on oeis.org

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

Pontus von Brömssen, Sep 08 2022

nonn

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.

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.

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

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))

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).

cp's OEIS Frontend

A233312 Terms of A114994 which are c-equivalent to "c-squares" (A020330).

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