A056734 Positive numbers k such that, in base 3, 2^k and 2^(k+1) have the same number of digits and the same number of 0's.

2, 5, 8, 10, 18, 21, 27, 29, 35, 40, 62, 67, 83, 92, 138, 146, 165, 184, 298, 346, 428, 487, 666, 750, 785, 929, 937, 1064, 1086, 1156, 1162, 1240, 1614, 1706, 1739, 1788, 2327, 2389, 2533, 2649, 2937, 3240, 3403, 3489, 3549, 3619, 3693, 3817, 3866, 4175

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 = 0.9513) with f(k) = c*k^(1/2) with c approximately 0.73. In addition, f'(k) approximates the probability that any particular k has this property. Any terms in A056154 must also be in this sequence.

Examples

			First term: 2^2 = 11_3, 2^3 = 22_3, both with 0 zeros and both of length 2.
Second term: 2^5 = 1012_3, 2^6 = 2101_3, both with 1 zero and both of length 4.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A056154, A117630.

Programs

Mathematica

Select[Range[4200],IntegerLength[2^#,3]==IntegerLength[2^(#+1),3] && DigitCount[ 2^#,3,0]==DigitCount[2^(#+1),3,0]&] (* Harvey P. Dale, Dec 10 2021 *)

PARI

isok(k) = my(da=digits(2^k, 3), db=digits(2^(k+1), 3)); (#da == #db) && (#select(x->(x==0), da) == #select(x->(x==0), db)); \\ Michel Marcus, Jul 01 2021