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A056735
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn,base
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AUTHOR
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Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000
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STATUS
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approved
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