A244041 Sum of digits of n written in fractional base 4/3.

0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 6, 7, 8, 9, 9, 10, 11, 12, 8, 9, 10, 11, 10, 11, 12, 13, 8, 9, 10, 11, 11, 12, 13, 14, 12, 13, 14, 15, 9, 10, 11, 12, 11, 12, 13, 14, 14, 15, 16, 17, 14, 15, 16, 17, 10, 11, 12, 13, 11, 12, 13, 14, 14, 15, 16, 17

Offset: 0

Author

Hailey R. Olafson, Jun 17 2014

Keywords

• nonn
• base

Comments

The base 4/3 expansion is unique and thus the sum of digits function is well-defined.

Examples

			In base 4/3 the number 14 is represented by 3212 and so a(14) = 3 + 2 + 1 + 2 = 8.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
• Kevin Ryde, Plot for Upper/Lower Bound Factors (and LaTeX source).
• Index entries for sequences related to fractional bases

Crossrefs

Cf. A000120, A007953, A024631, A053737, A244040.

Programs

Mathematica

p:=4; q:=3; a[n_]:= a[n]= If[n==0, 0, a[q*Floor[n/p]] + Mod[n, p]]; Table[a[n], {n,0,75}] (* G. C. Greubel, Aug 20 2019 *)

PARI

a(n) = p=4; q=3; if(n==0,0, a(q*(n\p)) + (n%p));
vector(75, n, n--; a(n)) \\ G. C. Greubel, Aug 20 2019

Sage

def base43sum(n):
    L, i = [n], 1
    while L[i-1]>3:
        x=L[i-1]
        L[i-1]=x.mod(4)
        L.append(3*floor(x/4))
        i+=1
    return sum(L)
[base43sum(n) for n in [0..75]]

Formula

a(n) = A007953(A024631(n)). - Michel Marcus, Jun 17 2014

a(n) < 3 log(n)/log(4/3) < 11 log(n) for n > 1. Possibly the constant factor can be replaced by 7 or 8. - Charles R Greathouse IV, Sep 22 2022

Conjecture: a(n) >> log(n), hence a(n) ≍ log(n). - Charles R Greathouse IV, Nov 03 2022