cp's OEIS Frontend

A121016 Numbers whose binary expansion is properly periodic.

%I A121016 #10 Oct 31 2019 14:32:32
%S A121016 3,7,10,15,31,36,42,45,54,63,127,136,153,170,187,204,221,238,255,292,
%T A121016 365,438,511,528,561,594,627,660,682,693,726,759,792,825,858,891,924,
%U A121016 957,990,1023,2047,2080,2145,2184,2210,2275,2340,2405,2457,2470,2535
%N A121016 Numbers whose binary expansion is properly periodic.
%C A121016 A finite sequence is aperiodic if its cyclic rotations are all different. - _Gus Wiseman_, Oct 31 2019
%H A121016 Charles R Greathouse IV, <a href="/A121016/b121016.txt">Table of n, a(n) for n = 1..10000</a>
%e A121016 For example, 204=(1100 1100)_2 and 292=(100 100 100)_2 belong to the sequence, but 30=(11110)_2 cannot be split into repeating periods.
%e A121016 From _Gus Wiseman_, Oct 31 2019: (Start)
%e A121016 The sequence of terms together with their binary expansions and binary indices begins:
%e A121016    3:         11 ~ {1,2}
%e A121016    7:        111 ~ {1,2,3}
%e A121016    10:      1010 ~ {2,4}
%e A121016    15:      1111 ~ {1,2,3,4}
%e A121016    31:     11111 ~ {1,2,3,4,5}
%e A121016    36:    100100 ~ {3,6}
%e A121016    42:    101010 ~ {2,4,6}
%e A121016    45:    101101 ~ {1,3,4,6}
%e A121016    54:    110110 ~ {2,3,5,6}
%e A121016    63:    111111 ~ {1,2,3,4,5,6}
%e A121016   127:   1111111 ~ {1,2,3,4,5,6,7}
%e A121016   136:  10001000 ~ {4,8}
%e A121016   153:  10011001 ~ {1,4,5,8}
%e A121016   170:  10101010 ~ {2,4,6,8}
%e A121016   187:  10111011 ~ {1,2,4,5,6,8}
%e A121016   204:  11001100 ~ {3,4,7,8}
%e A121016   221:  11011101 ~ {1,3,4,5,7,8}
%e A121016   238:  11101110 ~ {2,3,4,6,7,8}
%e A121016   255:  11111111 ~ {1,2,3,4,5,6,7,8}
%e A121016   292: 100100100 ~ {3,6,9}
%e A121016 (End)
%t A121016 PeriodicQ[n_, base_] := Block[{l = IntegerDigits[n, base]}, MemberQ[ RotateLeft[l, # ] & /@ Most@ Divisors@ Length@l, l]]; Select[ Range@2599, PeriodicQ[ #, 2] &]
%o A121016 (PARI) is(n)=n=binary(n);fordiv(#n,d,for(i=1,#n/d-1, for(j=1,d, if(n[j]!=n[j+i*d], next(3)))); return(d<#n)) \\ _Charles R Greathouse IV_, Dec 10 2013
%Y A121016 A020330 is a subsequence.
%Y A121016 Numbers whose binary expansion is aperiodic are A328594.
%Y A121016 Numbers whose reversed binary expansion is Lyndon are A328596.
%Y A121016 Numbers whose binary indices have equal run-lengths are A164707.
%Y A121016 Cf. A000120, A003714, A014081, A065609, A069010, A275692, A328595.
%K A121016 base,easy,nonn
%O A121016 1,1
%A A121016 _Jacob A. Siehler_, Sep 08 2006