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Number k such that k and k+1 are both digitally balanced numbers in base 2 (A031443).
5

%I #29 Nov 23 2020 02:05:40

%S 9,37,41,49,141,149,153,165,169,177,197,201,209,225,541,557,565,569,

%T 589,597,601,613,617,625,653,661,665,677,681,689,709,713,721,737,781,

%U 789,793,805,809,817,837,841,849,865,901,905,913,929,961,2109,2141,2157,2165

%N Number k such that k and k+1 are both digitally balanced numbers in base 2 (A031443).

%C All the terms are of the form 4*k + 1, where k is a digitally balanced number in base 2. Therefore, there are no 3 consecutive numbers that are digitally balanced in base 2.

%C The number of terms below 2^k is A079309(floor(k/2)-1) for k > 3.

%H Amiram Eldar, <a href="/A337238/b337238.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 4*A031443(n) + 1.

%e 9 is a term since the binary representation of 9 is 1001, which contains 2 0's and 2 1's, and the binary representation of 9 + 1 = 10 is 1010, which also contains 2 0's and 2 1's.

%t digBalQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ@(m = Length@d) && Count[d, 1] == m/2]; Select[Range[2000], digBalQ[#] && digBalQ[# + 1] &]

%Y Cf. A031443, A079309, A191292.

%Y A206374 \ {2} is a subsequence.

%K nonn,base,easy

%O 1,1

%A _Amiram Eldar_, Nov 21 2020