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%I #20 Apr 04 2021 13:04:03
%S 5,27,40,92,138,929,1086,352664,4976816,9914261,23434996,30490425,
%T 49094174
%N Numbers n such that the number of times each digit occurs in 2^n, represented in base 3, is the same as 2^(n+1), also represented in base 3. Or in other words, when represented in base 3, the digits in 2^n can be rearranged to form 2^(n+1).
%C For powers of 2 less than 2^1000, representations in base 3 are the only nontrivial examples where these kinds of pairs can be found. In other bases, for any integer n > 1, 2^(n+2) has the same frequency of digits as 2^(2n), represented in base (2^n)+1 (e.g., 2^3 and 2^4 in base 5, 2^4 and 2^6 in base 9, 2^5 and 2^8 in base 17, etc.).
%C For any n > 0, it can be shown that the distribution of these terms is approximately k*log(n), with k a small constant. This distribution can be derived from empirical evidence detailed in sequences A056734, A056735 and A056736.
%H J. Frech, <a href="https://papers.jfrech.com/2019-10-28_jonathan-frech_extending-a056154.pdf">Extending A056154</a>, 2019.
%e First term: 2^5 = 1012 and 2^6 = 2101 -> number of occurrences of 0, 1 and 2 are {1 2 1}; second term: 2^27 = 100100112222002222 and 2^28 = 200201002221012221 -> {6 4 8}.
%Y Cf. A056734, A056735, A056736.
%K hard,more,nonn,base
%O 1,1
%A Russell Harper (rharper(AT)intouchsurvey.com), Jul 30 2000
%E More terms from Bruce G. Stewart (bstewart(AT)bix.com), Aug 28 2000 and Sep 15 2000
%E a(13) from _Jonathan Frech_, Oct 31 2019
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