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

1,1

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.

Links

Table of n, a(n) for n=1..47.

Example

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.

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 *)

Crossrefs

Cf. A007089, A056154.

Sequence in context: A091721 A039283 A045162 * A298441 A056154 A156215

Adjacent sequences: A056732 A056733 A056734 * A056736 A056737 A056738

Keyword

easy,nonn,base

Author

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

Status

approved