login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A073138 Largest number having in its binary representation the same number of 0's and 1's as n. 19
0, 1, 2, 3, 4, 6, 6, 7, 8, 12, 12, 14, 12, 14, 14, 15, 16, 24, 24, 28, 24, 28, 28, 30, 24, 28, 28, 30, 28, 30, 30, 31, 32, 48, 48, 56, 48, 56, 56, 60, 48, 56, 56, 60, 56, 60, 60, 62, 48, 56, 56, 60, 56, 60, 60, 62, 56, 60, 60, 62, 60, 62, 62, 63, 64, 96, 96, 112, 96, 112, 112 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A023416(a(n))=A023416(n), A000120(a(n))=A000120(n).

Comment from Trevor Green (green(AT)snoopy.usask.ca), Nov 26 2003: (Start)

a(n)/n has an accumulation point at x exactly when x is in the interval [1, 2]. Proof: Clearly n <= a(n) < 2n. Let b(n) = a(n)/n, then b(n) must always lie in [1,2) and all the accumulation points of the sequence must lie in [1,2]. We shall show that every such number is an accumulation point.

First, consider any d-bit integer n. Suppose that z of these bits are 0. Let n' be the (d+z)-bit integer whose first d bits are the same as those of n and whose remaining bits are all 1. Then a(n') will have to be the (d+z)-bit integer whose first d bits are all 1 and whose last z bits are all 0.

Thus n' = (n+1)*2^z-1; a(n') = (2^d-1)2^z; and b(n') = (2^d-1)/(n+1) + epsilon, where 0 < epsilon < 2^(1-d). So to get an accumulation point x, we just choose n(d) to be the d-bit integer such that (2^d-1)/(n(d)+1) < x <= (2^d-1)/n(d), or equivalently, n(d) = floor((2^d-1)/x). If x lies in [1,2), then n(d) will always be a d-bit number for sufficiently large d.

Then n'(d) yields an increasing subsequence of the integers for which b(n'(d)) converges to x. For x = 2, choose n(d) = 2^(d-1), which is always a d-bit number; then b(n'(d)) = (2^d-1)/(2^(d-1)+1) + epsilon = 2 + epsilon', where epsilon' also heads for 0 as d blows up. This proves the claim.

(End)

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..8191 (terms 0..1023 from T. D. Noe)

Index entries for sequences related to binary expansion of n

FORMULA

a(n+1) = a(floor(n/2))*2 + (n mod 2)*(2^log2(n) - a(floor(n/2))); a(0)=0.

a(0)=0, a(1)=1, a(2n) = 2a(n), a(2n+1) = a(n) + 2^[log2(n)]. - Ralf Stephan, Oct 05 2003

a(n) = 2^(Floor[Log[2, n]] + 1) * (1 - 2^(-d(n))) d(n) = digit sum of base 2 expansion of n. - Trevor G. Hyde (thyde12(AT)amherst.edu), Jul 14 2008

a(n) = A038573(n) * A080100(n). - Reinhard Zumkeller, Jan 16 2012

EXAMPLE

a(20)=24, as 20='10100' and 24 is the greatest number having two 1's and three 0's: 17='10001', 18='10010', 20='10100' and 24='11000'.

MATHEMATICA

f[n_] := Module[{idn=IntegerDigits[n, 2], o, l}, l=Length[idn]; o=Count[idn, 1]; FromDigits[Join[Table[1, {o}], Table[0, {l-o}]], 2]]; Table[f[i], {i, 0, 70}]

ln[n_] := Module[{idn=IntegerDigits[n, 2], len, zer}, len=Length[idn]; zer=Count[idn, 0]; FromDigits[Join[Table[1, {len-zer}], Table[0, {zer}]], 2]]; Table[ln[i], {i, 0, 70}]

a[z_] := 2^(Floor[Log[2, z]] + 1) * (1 - 2^(-Sum[k, {k, IntegerDigits[n, 2]}])) Column[Table[a[p], {p, 500}], Right] (* Trevor G. Hyde (thyde12(AT)amherst.edu), Jul 14 2008 *)

PROG

(Haskell)

a073138 n = a038573 n * a080100 n  -- Reinhard Zumkeller, Jan 16 2012

(PARI) a(n) = fromdigits(vecsort(binary(n), , 4), 2); \\ Michel Marcus, Sep 26 2018

CROSSREFS

Cf. A007088, A073137, A000523, A073139, A073140, A073141, A319650.

Cf. A030109.

Cf. A038573.

Decimal equivalent of A221714. - N. J. A. Sloane, Jan 26 2013

Sequence in context: A246593 A256999 A331857 * A342179 A214965 A134361

Adjacent sequences:  A073135 A073136 A073137 * A073139 A073140 A073141

KEYWORD

nonn,nice,look

AUTHOR

Reinhard Zumkeller, Jul 16 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 11 14:29 EDT 2021. Contains 342886 sequences. (Running on oeis4.)