A096288 Sum of digits of n in bases 2 and 3.

0, 2, 3, 3, 3, 5, 4, 6, 5, 3, 4, 6, 4, 6, 7, 7, 5, 7, 4, 6, 6, 6, 7, 9, 6, 8, 9, 5, 5, 7, 6, 8, 5, 5, 6, 8, 4, 6, 7, 7, 6, 8, 7, 9, 9, 7, 8, 10, 6, 8, 9, 9, 9, 11, 6, 8, 7, 7, 8, 10, 8, 10, 11, 9, 5, 7, 6, 8, 8, 8, 9, 11, 6, 8, 9, 9, 9, 11, 10, 12, 10, 4, 5, 7, 5, 7, 8, 8, 7, 9, 6, 8, 8, 8, 9, 11, 6, 8, 9, 7

Offset: 0

Author

Miklos Kristof, Peter Boros, Jun 24 2004

Keywords

• easy
• nonn
• base

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

Examples

			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.
		

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.

Crossrefs

Cf. A000120, A053735, A095190, A010059, A010060.

Programs

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 *)

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

Formula

a(n) = A000120(n) + A053735(n). - Amiram Eldar, Jul 28 2023

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

Extensions

Edited by N. J. A. Sloane, Dec 05 2019