A244041 - OEIS

%I #32 Mar 26 2024 09:14:38

%S 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,

%T 12,13,8,9,10,11,11,12,13,14,12,13,14,15,9,10,11,12,11,12,13,14,14,15,

%U 16,17,14,15,16,17,10,11,12,13,11,12,13,14,14,15,16,17

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

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

%H G. C. Greubel, <a href="/A244041/b244041.txt">Table of n, a(n) for n = 0..1000</a>

%H F. M. Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Dekking/dek25.html">The Thue-Morse Sequence in Base 3/2</a>, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.

%H Kevin Ryde, <a href="/A244041/a244041_1.pdf">Plot for Upper/Lower Bound Factors</a> (and <a href="/A244041/a244041.tex">LaTeX source</a>).

%H <a href="/index/Ba#base_fractional">Index entries for sequences related to fractional bases</a>

%F a(n) = A007953(A024631(n)). - _Michel Marcus_, Jun 17 2014

%F 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

%F Conjecture: a(n) >> log(n), hence a(n) ≍ log(n). - _Charles R Greathouse IV_, Nov 03 2022

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

%t 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 *)

%o (Sage)

%o def base43sum(n):

%o     L, i = [n], 1

%o     while L[i-1]>3:

%o         x=L[i-1]

%o         L[i-1]=x.mod(4)

%o         L.append(3*floor(x/4))

%o         i+=1

%o     return sum(L)

%o [base43sum(n) for n in [0..75]]

%o (PARI) a(n) = p=4; q=3; if(n==0,0, a(q*(n\p)) + (n%p));

%o vector(75, n, n--; a(n)) \\ _G. C. Greubel_, Aug 20 2019

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

%K nonn,base

%O 0,3

%A _Hailey R. Olafson_, Jun 17 2014