A056850 Minimal absolute difference of 3^n and 2^k.

0, 1, 1, 5, 17, 13, 217, 139, 1631, 3299, 6487, 46075, 7153, 502829, 588665, 2428309, 9492289, 5077565, 118985033, 88519643, 808182895, 1870418611, 2978678759, 25423702091, 7551629537, 252223018333, 342842572777, 1170495537221, 5284606410545, 1738366812781

Offset: 0

Author

Robert G. Wilson v, Aug 30 2000

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A056577 (smallest 3^n-2^k), A063003 (smallest 2^k-3^n).

Programs

Mathematica

Table[Min[# - 2^Floor@ Log2@ # &[3^n], 2^Ceiling@ Log2@ # - # &[3^n]], {n, 0, 27}]

Extensions

a(28)-a(29) from Jon E. Schoenfield, Mar 31 2021