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A272614 Numbers whose binary digits, except for the first "1", are given by floor(((k-n)/n) mod 2) with 1<=k<=n.

%I A272614 #35 May 25 2021 07:58:26
%S A272614 1,2,6,8,28,40,104,144,496,672,1632,2240,7872,11648,27520,33536,
%T A272614 120576,175616,445952,629760,2014208,2701312,6453248,8712192,33353728,
%U A272614 48881664,114548736,144949248,476561408,684687360,1787789312,2501836800,8510177280,11647451136,27590000640
%N A272614 Numbers whose binary digits, except for the first "1", are given by floor(((k-n)/n) mod 2) with 1<=k<=n.
%C A272614 Numbers such that the sequence of its binary digits change periodically with linearly increasing period depending of its position: after the first '1', k-th most significant bit changes with period 2*(k-1). For instance, the second most significant bit changes with period 2, third bit changes with period 4, fourth bit with period 6, and so on. For example on the first 15 terms we have:
%C A272614 a(n)      Binary digits
%C A272614 1         1
%C A272614 2         1 0
%C A272614 6         1 1 0
%C A272614 8         1 0 0 0
%C A272614 28        1 1 1 0 0
%C A272614 40        1 0 1 0 0 0
%C A272614 104       1 1 0 1 0 0 0
%C A272614 144       1 0 0 1 0 0 0 0
%C A272614 496       1 1 1 1 1 0 0 0 0
%C A272614 672       1 0 1 0 1 0 0 0 0 0
%C A272614 1632      1 1 0 0 1 1 0 0 0 0 0
%C A272614 2240      1 0 0 0 1 1 0 0 0 0 0 0
%C A272614 7872      1 1 1 1 0 1 1 0 0 0 0 0 0
%C A272614 11648     1 0 1 1 0 1 1 0 0 0 0 0 0 0
%C A272614 27520     1 1 0 1 0 1 1 1 0 0 0 0 0 0 0
%C A272614          /  | | | | |  \          etc.
%C A272614       MSB  /  | | | |   Period 12
%C A272614           /   | | |  \
%C A272614   Period 2   /  | |   Period 10
%C A272614             /   |  \
%C A272614     Period 4    |   Period 8
%C A272614                /
%C A272614        Period 6
%C A272614 Regarding the periodicity of the binary digits, this sequence is similar to A059893 where the periodicity of its binary digits are powers of two.
%C A272614 By truncating the least significant bits in such a way to leave only k most significant bits of a(n) with n>k-1, it is obtained a periodic sequence with period p given by p=Least common multiple (LCM) of {2,4,6,..,2k}. In general any subsequence obtained by a selection of a subset of its most significant bits including the most significant bit is periodic.
%F A272614 a(n) = 2^n + Sum_{k=1..n}[floor(((n-k)/k) mod 2) * 2^(n-k)].
%t A272614 nmax = 34;
%t A272614 a[n_] := 2^n + Sum[ Floor@Mod[(n - k)/k, 2]* 2^(n - k), {k, 1, n}];
%t A272614 Table[a[n] , {n, 0, nmax}]
%o A272614 (PARI) a(n) = 2^n + sum(k=1, n, (floor(((n-k)/k)) % 2) * 2^(n-k)); \\ _Michel Marcus_, May 20 2016
%Y A272614 Cf. A059893, A272170, A271591.
%K A272614 nonn,base
%O A272614 0,2
%A A272614 _Andres Cicuttin_, May 03 2016