OFFSET
0,2
COMMENTS
Let n = Sum(c(k)*2^k), c(k) = 0,1, be the binary form of n, n = Sum(d(k)*3^k), d(k) = 0,1,2, the ternary form; then a(n) = Sum(c(k)+d(k)).
a(n) mod 2 = doubled Thue-Morse sequence A095190.
Let s[b](n) denote the sum of the digits of n to the base b. Senge and Straus proved in 1973 that s[a](n) + s[b](n) approaches infinity as n approaches infinity if and only if log(a)/log(b) is irrational. Stewart (1980) obtained an effectively computable lower bound. - David Radcliffe, Jan 16 2024
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram, and Bernard Landreau, Sums of the digits in bases 2 and 3, in: C. Elsholtz and P. Grabner (eds.), Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday, Springer, Cham, 2017, pp. 211-217; arXiv preprint, arXiv:1611.08180 [math.NT], 2016.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Period Math Hung 3 (1973), 93-100.
Cameron Stewart, On the representation of an integer in two different bases, Journal für die reine und angewandte Mathematik, 319 (1980), 63-72.
FORMULA
a(n) > (log log n) / (log log log n + C) - 1 for n > 25, where C is effectively computable (Stewart 1980). - David Radcliffe, Jan 16 2024
EXAMPLE
n=11: 11=1*2^3+1*2^1+1*2^0, 1+1+1=3, 11=1*3^2+2*3^0, 1+2=3, so a(11)=3+3=6.
MAPLE
f := proc(n) local t1, t2, i;
t1:=convert(n, base, 2); t2:=convert(n, base, 3);
add(t1[i], i=1..nops(t1))+ add(t2[i], i=1..nops(t2));
end; # N. J. A. Sloane, Dec 05 2019
MATHEMATICA
a[n_] := Total @ IntegerDigits[n, 2] + Total @ IntegerDigits[n, 3];
a /@ Range[0, 100] (* Jean-François Alcover, Aug 21 2020 *)
Table[Total[Flatten[IntegerDigits[n, {2, 3}]]], {n, 0, 100}] (* Harvey P. Dale, Jan 29 2021 *)
PROG
(PARI) a(n) = sumdigits(n, 2) + sumdigits(n, 3); \\ Michel Marcus, Aug 21 2020
(Python)
sumdigits = lambda n, b: n % b + sumdigits(n // b, b) if n else 0
a = lambda n: sumdigits(n, 2) + sumdigits(n, 3) # David Radcliffe, Jan 16 2024
CROSSREFS
KEYWORD
easy,nonn,base
AUTHOR
Miklos Kristof, Peter Boros, Jun 24 2004
EXTENSIONS
Edited by N. J. A. Sloane, Dec 05 2019
STATUS
approved