

A261751


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


2



0, 1, 2, 3, 4, 6, 8, 16, 23, 32, 64, 91, 128, 256, 512, 1024, 2048, 4096, 5793, 8192, 16384, 32768, 46341, 65536, 92682, 131072, 185364, 262144, 370728, 524288, 1048576, 2097152, 2965821, 4194304, 5931642, 8388608, 16777216, 33554432, 47453133, 67108864, 94906266
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OFFSET

1,3


COMMENTS

2^k is always a term in this sequence.
It appears that all solutions are either a power of 2 or approximately sqrt(2) * a power of 2.  Andrew Howroyd, Dec 24 2019


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000


EXAMPLE

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


MATHEMATICA

SetBeginSet[set1_, set2_] :=
Do[For[i = 1, i <= Length[set1], i++, If[! set1[[i]] == set2[[i]], Return[False]]]; Return[True], {1}];
For[k = 0; set = {}, k <= 100000, k++, If[SetBeginSet[IntegerDigits[k, 2], IntegerDigits[k^3, 2]], Print[k]]]


PROG

(PARI) ok(n)={my(t=n^3); t == 0  t>>(logint(t, 2)logint(n, 2))==n} \\ Andrew Howroyd, Dec 23 2019
(PARI) \\ for larger values
viable(b, k)={my(p=b^3, q=(b+2^k1)^3, s=logint(q, 2), t=slogint(b, 2)+k); (p>>s)==0  ((p>>t)<=(b>>k) && (b>>k)<=(q>>t))}
upto(n)={
local(L=List([0]));
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))));
for(k=0, logint(n, 2), recurse(2^k, k));
Vec(L);
} \\ Andrew Howroyd, Dec 24 2019


CROSSREFS

Base 2 version of A052210.
Cf. A004539.
Sequence in context: A000031 A298072 A111023 * A294679 A333160 A345250
Adjacent sequences: A261748 A261749 A261750 * A261752 A261753 A261754


KEYWORD

nonn,base,easy


AUTHOR

Dhilan Lahoti, Aug 30 2015


EXTENSIONS

Terms a(31) and beyond from Andrew Howroyd, Dec 23 2019


STATUS

approved



