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

%I A244041 #32 Mar 26 2024 09:14:38
%S A244041 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 A244041 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 A244041 16,17,14,15,16,17,10,11,12,13,11,12,13,14,14,15,16,17
%N A244041 Sum of digits of n written in fractional base 4/3.
%C A244041 The base 4/3 expansion is unique and thus the sum of digits function is well-defined.
%H A244041 G. C. Greubel, <a href="/A244041/b244041.txt">Table of n, a(n) for n = 0..1000</a>
%H A244041 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 A244041 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 A244041 <a href="/index/Ba#base_fractional">Index entries for sequences related to fractional bases</a>
%F A244041 a(n) = A007953(A024631(n)). - _Michel Marcus_, Jun 17 2014
%F A244041 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 A244041 Conjecture: a(n) >> log(n), hence a(n) ≍ log(n). - _Charles R Greathouse IV_, Nov 03 2022
%e A244041 In base 4/3 the number 14 is represented by 3212 and so a(14) = 3 + 2 + 1 + 2 = 8.
%t A244041 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 A244041 (Sage)
%o A244041 def base43sum(n):
%o A244041     L, i = [n], 1
%o A244041     while L[i-1]>3:
%o A244041         x=L[i-1]
%o A244041         L[i-1]=x.mod(4)
%o A244041         L.append(3*floor(x/4))
%o A244041         i+=1
%o A244041     return sum(L)
%o A244041 [base43sum(n) for n in [0..75]]
%o A244041 (PARI) a(n) = p=4; q=3; if(n==0,0, a(q*(n\p)) + (n%p));
%o A244041 vector(75, n, n--; a(n)) \\ _G. C. Greubel_, Aug 20 2019
%Y A244041 Cf. A000120, A007953, A024631, A053737, A244040.
%K A244041 nonn,base
%O A244041 0,3
%A A244041 _Hailey R. Olafson_, Jun 17 2014