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A121016
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Numbers whose binary expansion is properly periodic.
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30
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3, 7, 10, 15, 31, 36, 42, 45, 54, 63, 127, 136, 153, 170, 187, 204, 221, 238, 255, 292, 365, 438, 511, 528, 561, 594, 627, 660, 682, 693, 726, 759, 792, 825, 858, 891, 924, 957, 990, 1023, 2047, 2080, 2145, 2184, 2210, 2275, 2340, 2405, 2457, 2470, 2535
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OFFSET
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1,1
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COMMENTS
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A finite sequence is aperiodic if its cyclic rotations are all different. - Gus Wiseman, Oct 31 2019
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LINKS
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EXAMPLE
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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.
The sequence of terms together with their binary expansions and binary indices begins:
3: 11 ~ {1,2}
7: 111 ~ {1,2,3}
10: 1010 ~ {2,4}
15: 1111 ~ {1,2,3,4}
31: 11111 ~ {1,2,3,4,5}
36: 100100 ~ {3,6}
42: 101010 ~ {2,4,6}
45: 101101 ~ {1,3,4,6}
54: 110110 ~ {2,3,5,6}
63: 111111 ~ {1,2,3,4,5,6}
127: 1111111 ~ {1,2,3,4,5,6,7}
136: 10001000 ~ {4,8}
153: 10011001 ~ {1,4,5,8}
170: 10101010 ~ {2,4,6,8}
187: 10111011 ~ {1,2,4,5,6,8}
204: 11001100 ~ {3,4,7,8}
221: 11011101 ~ {1,3,4,5,7,8}
238: 11101110 ~ {2,3,4,6,7,8}
255: 11111111 ~ {1,2,3,4,5,6,7,8}
292: 100100100 ~ {3,6,9}
(End)
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MATHEMATICA
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PeriodicQ[n_, base_] := Block[{l = IntegerDigits[n, base]}, MemberQ[ RotateLeft[l, # ] & /@ Most@ Divisors@ Length@l, l]]; Select[ Range@2599, PeriodicQ[ #, 2] &]
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PROG
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(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
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CROSSREFS
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Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary indices have equal run-lengths are A164707.
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KEYWORD
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base,easy,nonn
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AUTHOR
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STATUS
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approved
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