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
Keywords
Examples
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}
Crossrefs
Programs
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Mathematica
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1]; Select[Range[1000],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]
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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
Extensions
a(33)-a(40) from Andrew Howroyd, Sep 29 2019