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%I #12 Aug 14 2021 02:16:11
%S 5,27,32,40,54,92,135,138,151,159,167,176,189,281,284,319,401,503,718,
%T 723,734,820,929,1035,1086,1127,1311,1341,1371,1693,1785,1869,1948,
%U 2010,2181,2408,2563,2771,2923,2983,3004,3007,3210,3213,3479,3527,4037
%N 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.
%C 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.
%e a(1)=5: 2^5 = 1012_3, 2^6 = 2101_2, both with two 1's and both of length 4.
%e a(2)=27: 2^27 = 100100112222002222_3, 2^28 = 200201002221012221_3, both with four 1's and both of length 18.
%t 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 *)
%Y Cf. A007089, A056154.
%K easy,nonn,base
%O 1,1
%A Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000