A164707 A positive integer n is included if all runs of 1's in binary n are of the same length.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 27, 28, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 48, 51, 54, 56, 60, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 96, 99, 102, 108, 112, 119, 120, 124, 126, 127, 128, 129, 130, 132, 133, 136

Offset: 1

Author

Leroy Quet, Aug 23 2009

Keywords

• base
• nonn

Comments

Clarification: A binary number consists of "runs" completely of 1's alternating with runs completely of 0's. No two or more runs all of the same digit are adjacent.

This sequence contains in part positive integers that each contain one run of 1's. For those members of this sequence each with at least two runs of 1's, see A164709.

Examples

			From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   4:    100 ~ {3}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
   8:   1000 ~ {4}
   9:   1001 ~ {1,4}
  10:   1010 ~ {2,4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  16:  10000 ~ {5}
  17:  10001 ~ {1,5}
  18:  10010 ~ {2,5}
  20:  10100 ~ {3,5}
  21:  10101 ~ {1,3,5}
  24:  11000 ~ {4,5}
  27:  11011 ~ {1,2,4,5}
(End)
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A164708, A164709, A164710.

The version for prime indices is A072774.

The binary expansion of n has A069010(n) runs of 1's.

Numbers whose runs are all of different lengths are A328592.

Partitions with equal multiplicities are A047966.

Numbers whose binary expansion is aperiodic are A328594.

Numbers whose reversed binary expansion is a necklace are A328595.

Numbers whose reversed binary expansion is a Lyndon word are A328596.

Cf. A000120, A003714, A014081, A065609, A070939, A121016, A245563, A275692.

Programs

Maple

isA164707 := proc(n) local bdg,arl,lset ; bdg := convert(n,base,2) ; lset := {} ; arl := -1 ; for p from 1 to nops(bdg) do if op(p,bdg) = 1 then if p = 1 then arl := 1 ; else arl := arl+1 ; end if; else if arl > 0 then lset := lset union {arl} ; end if; arl := 0 ; end if; end do ; if arl > 0 then lset := lset union {arl} ; end if; return (nops(lset) <= 1 ); end proc: for n from 1 to 300 do if isA164707(n) then printf("%d,",n) ; end if; end do; # R. J. Mathar, Feb 27 2010

Mathematica

Select[Range@ 140, SameQ @@ Map[Length, Select[Split@ IntegerDigits[#, 2], First@ # == 1 &]] &] (* Michael De Vlieger, Aug 20 2017 *)

Perl

foreach(1..140){
    %runs=();
    $runs{$}++ foreach split /0+/, sprintf("%b",$);
    print "$_, " if 1==keys(%runs);
}
# Ivan Neretin, Nov 09 2015

Extensions

Extended beyond 42 by R. J. Mathar, Feb 27 2010