A327368 The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.

1, 2, 4, 8, 16, 32, 64, 128, 130, 256, 257, 512, 1024, 2048, 2084, 2316, 4096, 8192, 16384, 32768, 32776, 32777, 65536, 131072, 131074, 131200, 131457, 131462, 133390, 165920, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 8388640, 8388897, 8390688, 8519840

Offset

1,2

Links

Table of n, a(n) for n=1..40.

Example

The sequence of terms together with their binary indices begins:
  2      {2}
  4      {3}
  8      {4}
  16     {5}
  32     {6}
  64     {7}
  128    {8}
  130    {2,8}
  256    {9}
  257    {1,9}
  512    {10}
  1024   {11}
  2048   {12}
  2084   {3,6,12}
  2316   {3,4,9,12}
  4096   {13}
  8192   {14}
  16384  {15}
  32768  {16}
  32776  {4,16}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[1000], IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

Prog

(PARI) ok(n)={my(s=0, p=1, k=0); for(i=0, logint(n, 2), if(bittest(n, i), s+=i+1; p*=i+1; k++)); s%k==0 && ispower(p, k)}
{ for(n=1, 10^7, if(ok(n), print1(n, ", "))) } \\ Andrew Howroyd, Sep 29 2019

Crossrefs

A superset of A327777.

Numbers whose binary indices have integer mean: A326669

Numbers whose binary indices have integer geometric mean: A326673

Cf. A000120, A029931, A048793, A070939, A096111, A291166, A295235, A326029, A326643, A326699/A326700, A327777.

Sequence in context: A036147 A036146 A036144 * A286660 A009641 A089889

Adjacent sequences: A327365 A327366 A327367 * A327369 A327370 A327371

Keyword

nonn

Author

Gus Wiseman, Sep 27 2019

Extensions

a(33)-a(40) from Andrew Howroyd, Sep 29 2019

Status

approved