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

%I A327368 #12 Sep 30 2019 12:15:01
%S A327368 1,2,4,8,16,32,64,128,130,256,257,512,1024,2048,2084,2316,4096,8192,
%T A327368 16384,32768,32776,32777,65536,131072,131074,131200,131457,131462,
%U A327368 133390,165920,262144,524288,1048576,2097152,4194304,8388608,8388640,8388897,8390688,8519840
%N A327368 The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.
%e A327368 The sequence of terms together with their binary indices begins:
%e A327368   2      {2}
%e A327368   4      {3}
%e A327368   8      {4}
%e A327368   16     {5}
%e A327368   32     {6}
%e A327368   64     {7}
%e A327368   128    {8}
%e A327368   130    {2,8}
%e A327368   256    {9}
%e A327368   257    {1,9}
%e A327368   512    {10}
%e A327368   1024   {11}
%e A327368   2048   {12}
%e A327368   2084   {3,6,12}
%e A327368   2316   {3,4,9,12}
%e A327368   4096   {13}
%e A327368   8192   {14}
%e A327368   16384  {15}
%e A327368   32768  {16}
%e A327368   32776  {4,16}
%t A327368 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327368 Select[Range[1000],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]
%o A327368 (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)}
%o A327368 { for(n=1, 10^7, if(ok(n), print1(n, ", "))) } \\ _Andrew Howroyd_, Sep 29 2019
%Y A327368 A superset of A327777.
%Y A327368 Numbers whose binary indices have integer mean: A326669
%Y A327368 Numbers whose binary indices have integer geometric mean: A326673
%Y A327368 Cf. A000120, A029931, A048793, A070939, A096111, A291166, A295235, A326029, A326643, A326699/A326700, A327777.
%K A327368 nonn
%O A327368 1,2
%A A327368 _Gus Wiseman_, Sep 27 2019
%E A327368 a(33)-a(40) from _Andrew Howroyd_, Sep 29 2019