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A261751 Numbers n with property that binary expansion of n^3 begins with the binary expansion of n. 2

%I

%S 0,1,2,3,4,6,8,16,23,32,64,91,128,256,512,1024,2048,4096,5793,8192,

%T 16384,32768,46341,65536,92682,131072,185364,262144,370728,524288,

%U 1048576,2097152,2965821,4194304,5931642,8388608,16777216,33554432,47453133,67108864,94906266

%N Numbers n with property that binary expansion of n^3 begins with the binary expansion of n.

%C 2^k is always a term in this sequence.

%C It appears that all solutions are either a power of 2 or approximately sqrt(2) * a power of 2. - _Andrew Howroyd_, Dec 24 2019

%H Andrew Howroyd, <a href="/A261751/b261751.txt">Table of n, a(n) for n = 1..1000</a>

%e 23 is a term of this sequence because its cube written in base 2 (10111110000111) starts with its representation in base 2 (10111).

%t SetBeginSet[set1_, set2_] :=

%t Do[For[i = 1, i <= Length[set1], i++,If[! set1[[i]] == set2[[i]], Return[False]]];Return[True], {1}];

%t For[k = 0; set = {}, k <= 100000, k++,If[SetBeginSet[IntegerDigits[k, 2], IntegerDigits[k^3, 2]],Print[k]]]

%o (PARI) ok(n)={my(t=n^3); t == 0 || t>>(logint(t,2)-logint(n,2))==n} \\ _Andrew Howroyd_, Dec 23 2019

%o (PARI) \\ for larger values

%o viable(b,k)={my(p=b^3, q=(b+2^k-1)^3, s=logint(q,2), t=s-logint(b,2)+k); (p>>s)==0 || ((p>>t)<=(b>>k) && (b>>k)<=(q>>t))}

%o upto(n)={

%o local(L=List([0]));

%o my(recurse(b,k)=; if(b <= n && viable(b,k), k--; if(k<0, listput(L, b), self()(b,k); self()(b+2^k,k))));

%o for(k=0, logint(n,2), recurse(2^k, k));

%o Vec(L);

%o } \\ _Andrew Howroyd_, Dec 24 2019

%Y Base 2 version of A052210.

%Y Cf. A004539.

%K nonn,base,easy

%O 1,3

%A _Dhilan Lahoti_, Aug 30 2015

%E Terms a(31) and beyond from _Andrew Howroyd_, Dec 23 2019

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Last modified February 7 12:44 EST 2023. Contains 360123 sequences. (Running on oeis4.)