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a(n) = the (decimal equivalent of the) smallest integer that can be made by rotating the binary digits of n any number of positions to the left or right. a(n) in binary may contain fewer digits, after leading 0's are removed, than n written in binary has.
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%I #20 Aug 05 2016 09:19:42

%S 1,1,3,1,3,3,7,1,3,5,7,3,7,7,15,1,3,5,7,5,11,11,15,3,7,11,15,7,15,15,

%T 31,1,3,5,7,9,11,13,15,5,13,21,23,11,27,23,31,3,7,11,15,13,23,27,31,7,

%U 15,23,31,15,31,31,63,1,3,5,7,9,11,13,15,9,19,21,23,19,27,29,31,5,13,21,29

%N a(n) = the (decimal equivalent of the) smallest integer that can be made by rotating the binary digits of n any number of positions to the left or right. a(n) in binary may contain fewer digits, after leading 0's are removed, than n written in binary has.

%C By rotating the binary digits of n, it is meant: Write n in binary without any leading 0's. To rotate this string to the right, say, by one position, first remove the rightmost digit and then append it on the left side of the remaining string. (So the least significant digit becomes the most significant digit.) Here, leading 0's are not removed after the first rotation, so that each binary string being rotated has the same number of binary digits as n has.

%H Alois P. Heinz, <a href="/A163381/b163381.txt">Table of n, a(n) for n = 1..8190</a>

%e 13 in binary is 1101. Rotating this just once to the right, we get 1110, 14 in decimal. If we rotate twice to the right, we would have had 0111 = 7 in decimal. Rotating 3 times, we end up with 1011, which is 11 in decimal. Rotating 4 times, we end up at the beginning with 1101 = 13. 7 is the smallest of these, so a(13) = 7.

%p a:= proc(n) local i, k, m, s;

%p k, m, s:= ilog2(n), n, n;

%p for i to k do m:= iquo(m, 2, 'd') +d*2^k; s:=s, m od;

%p min(s)

%p end:

%p seq(a(n), n=1..80); # _Alois P. Heinz_, May 24 2012

%t Table[Min[FromDigits[ #,2]&/@ NestList[RotateLeft,IntegerDigits[n,2], Floor[Log[2,n]]]],{n,255}] (* _Wouter Meeussen_, Jul 27 2009 *)

%o (PARI) a(n)=local(k=#binary(n),m=2^k,t=n*(m+1));vecmin(vector(k,x,(t\2^x)%m)) \\ _Hagen von Eitzen_, Jul 27 2009

%Y Cf. A163380, A163382.

%K base,nonn,look

%O 1,3

%A _Leroy Quet_, Jul 25 2009

%E More terms from _Hagen von Eitzen_ and _Wouter Meeussen_, Jul 27 2009