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A093476
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Index of occurrence of the first 0 bit in binary representation of 3^n.
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1
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2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 6, 2, 2, 3, 2, 4, 2, 2, 3, 2, 4, 2, 7, 2, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 6, 2, 2, 3, 2, 4, 2, 2, 3, 2, 4, 2, 7, 2, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2
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OFFSET
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2,1
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LINKS
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FORMULA
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It seems that Sum_{i=2..n} a(i) is asymptotic to c*n with c=2.7(8).....
a(n) = k if log_2(2 - 1/2^(k-2)) < frac(n*log_2(3)) < log_2(2 - 1/2^(k-1)). By the equidistribution theorem, this occurs with asymptotic density log_2(2-1/2^(k-1)) - log_2(2-1/2^(k-2)).
Thus c = Sum_{k>=2} k (log_2(2-1/2^(k-1)) - log_2(2 - 1/2^(k-2))) = 2 - Sum_{k>=2} log_2(1-1/2^k) = 2.791916824662... Note that A048651 is the decimal expansion of 2^(1-c). (End)
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EXAMPLE
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In binary, 3^5 = [1, 1, 1, 1, 0, 0, 1, 1] where the first 0 occurs at 5th place. Hence a(5)=5.
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MAPLE
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seq(ListTools:-Search(0, ListTools:-Reverse(convert(3^n, base, 2))), n=2..200); # Robert Israel, Nov 20 2017
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MATHEMATICA
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Array[FirstPosition[IntegerDigits[3^#, 2], 0][[1]] &, 105, 2] (* Michael De Vlieger, Nov 20 2017 *)
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PROG
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(PARI) a(n)=if(n<2, 0, s=1; while(component(binary(3^n), s)>0, s++); s)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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