OFFSET
0,4
COMMENTS
Except for 3^0 - 2^0, 3^1 - 2^1 and 3^2 - 2^3, there are no cases where the differences are less than 4.
It is known that a(n) tends to infinity as n tends to infinity. Indeed, Tijdeman showed that there exists an effectively computable constant c > 0 such that |2^x - 3^y| > 2^x/x^c. - Tomohiro Yamada, Sep 29 2017
Empirical observation: For at least values a(0) through a(6308), k-2 < n*log_2(3) < k+2. - Matthew Schuster, Mar 28 2021
For all n >= 0, the lower and upper limits on n*log_2(3) - k are log_2(3/4) = -0.4150374... and log_2(3/2) = 0.5849625..., respectively; i.e., 0 <= n*log_2(3) - k - log_2(3/4) < 1. - Jon E. Schoenfield, Apr 21 2021
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2097
R. Tijdeman, On integers with many small prime factors, Compos. Math. 26 (1973), 319--330.
EXAMPLE
For n = 4, the closest power of 2 to 3^n = 81 is 2^6 = 64, so a(4) = |3^4 - 2^6| = |81 - 64| = 17. - Jon E. Schoenfield, Sep 30 2017
MATHEMATICA
Table[Min[# - 2^Floor@ Log2@ # &[3^n], 2^Ceiling@ Log2@ # - # &[3^n]], {n, 0, 27}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Aug 30 2000
EXTENSIONS
a(28)-a(29) from Jon E. Schoenfield, Mar 31 2021
STATUS
approved