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

1,1

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.

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A056154, A117630.

Sequence in context: A169922 A157481 A100809 * A019995 A188802 A031141

Adjacent sequences: A056731 A056732 A056733 * A056735 A056736 A056737

Keyword

easy,nonn,base

Author

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

Status

approved