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

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

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^k-1)^3, s=logint(q, 2), t=s-logint(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