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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 27 15:44 EDT 2021. Contains 347691 sequences. (Running on oeis4.)