A056734 - OEIS

%I #19 Dec 10 2021 16:43:53

%S 2,5,8,10,18,21,27,29,35,40,62,67,83,92,138,146,165,184,298,346,428,

%T 487,666,750,785,929,937,1064,1086,1156,1162,1240,1614,1706,1739,1788,

%U 2327,2389,2533,2649,2937,3240,3403,3489,3549,3619,3693,3817,3866,4175

%N 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.

%C 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.

%H Harvey P. Dale, <a href="/A056734/b056734.txt">Table of n, a(n) for n = 1..200</a>

%e First term: 2^2 = 11_3, 2^3 = 22_3, both with 0 zeros and both of length 2.

%e Second term: 2^5 = 1012_3, 2^6 = 2101_3, both with 1 zero and both of length 4.

%t 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 *)

%o (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

%Y Cf. A056154, A117630.

%K easy,nonn,base

%O 1,1

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