A023717 Numbers with no 3's in base-4 expansion.

0, 1, 2, 4, 5, 6, 8, 9, 10, 16, 17, 18, 20, 21, 22, 24, 25, 26, 32, 33, 34, 36, 37, 38, 40, 41, 42, 64, 65, 66, 68, 69, 70, 72, 73, 74, 80, 81, 82, 84, 85, 86, 88, 89, 90, 96, 97, 98, 100, 101, 102, 104, 105, 106, 128, 129, 130, 132, 133, 134, 136, 137, 138

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• base
• easy

Comments

A032925 is the intersection of this sequence and A023705; cf. A179888. - Reinhard Zumkeller, Jul 31 2010

Fixed point of the morphism: 0-> 0,1,2; 1-> 4,5,6; 2-> 8,9,10; ...; n-> 4n,4n+1,4n+2. - Philippe Deléham, Oct 22 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A032925, A023705, A179888.

Programs

C

uint32_t a_next(uint32_t a_n) {
    uint32_t t = ((a_n ^ 0xaaaaaaaa) | 0x55555555) >> 1;
    return (a_n - t) & t;
} // Falk Hüffner, Jan 22 2022

Haskell

a023717 n = a023717_list !! (n-1)
a023717_list = filter f [0..] where
   f x = x < 3 || (q < 3 && f x') where (x', q) = divMod x 4
-- Reinhard Zumkeller, Apr 18 2015

Julia

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 4
    end
r end; [a(n) for n in 0:58] |> println # Peter Luschny, Jan 03 2021

Mathematica

Select[ Range[ 0, 140 ], (Count[ IntegerDigits[ #, 4 ], 3 ]==0)& ]

PARI

a(n)=if(n<1,0,if(n%3,a(n-1)+1,4*a(n/3)))

PARI

a(n)=if(n<1,0,4*a(floor(n/3))+n-3*floor(n/3))

Python

from gmpy2 import digits
def A023717(n): return int(digits(n,3),4) # Chai Wah Wu, May 06 2025

Formula

a(n) = Sum_{i=0..m} d(i)*4^i, where Sum_{i=0..m} d(i)*3^i is the base-3 representation of n. - Clark Kimberling

a(3n) = 4*a(n); a(3n+1) = 4*a(n)+1; a(3n+2) = 4*a(n)+2; a(n) = 4*a(floor(n/3)) + n - 3*floor(n/3). - Benoit Cloitre, Apr 27 2003

a(n) = Sum_{k>=0} A030341(n,k)*4^k. - Philippe Deléham, Oct 22 2011