A056735 Numbers k such that the base-3 expansions of 2^k and 2^(k+1) have the same number of 1's and the same number of digits.

5, 27, 32, 40, 54, 92, 135, 138, 151, 159, 167, 176, 189, 281, 284, 319, 401, 503, 718, 723, 734, 820, 929, 1035, 1086, 1127, 1311, 1341, 1371, 1693, 1785, 1869, 1948, 2010, 2181, 2408, 2563, 2771, 2923, 2983, 3004, 3007, 3210, 3213, 3479, 3527, 4037

Offset: 1

Author

Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000

Keywords

• easy
• nonn
• base

Comments

Using empirical data for 1 <= k <= 10000, it has been found that the distribution of these terms correlates well (R^2 = 0.9798) with g(k) = b*sqrt(k) where b ~ 0.70. In addition, g'(k) approximates the probability that any particular k has this property. A056154 is a subsequence.

Examples

			a(1)=5: 2^5 = 1012_3, 2^6 = 2101_2, both with two 1's and both of length 4.
a(2)=27: 2^27 = 100100112222002222_3, 2^28 = 200201002221012221_3, both with four 1's and both of length 18.
		

Crossrefs

Cf. A007089, A056154.

Programs

Mathematica

Select[Range[4100],Length[IntegerDigits[2^#,3]]==Length[ IntegerDigits[ 2^(#+1),3]] && DigitCount[2^#,3,1]==DigitCount[2^(#+1),3,1]&] (* Harvey P. Dale, Jul 09 2021 *)