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A096288
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Sum of digits of n in bases 2 and 3.
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4
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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));
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MATHEMATICA
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a[n_] := Total @ IntegerDigits[n, 2] + Total @ IntegerDigits[n, 3];
Table[Total[Flatten[IntegerDigits[n, {2, 3}]]], {n, 0, 100}] (* Harvey P. Dale, Jan 29 2021 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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