cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A065280 The bisection of odd terms (the <= 0 half of Z) of A065279.

Original entry on oeis.org

1, 1, 1, 6, 2, 2, 12, 14, 4, 4, 4, 4, 24, 26, 28, 30, 8, 8, 8, 8, 8, 8, 8, 8, 48, 50, 52, 54, 56, 58, 60, 62, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 32, 32, 32, 32, 32, 32
Offset: 1

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Author

Antti Karttunen, Oct 28 2001

Keywords

A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64
Offset: 0

Views

Author

Henry Bottomley, Mar 22 2000

Keywords

Comments

Except for the initial term, 2^n appears 2^n times. - Lekraj Beedassy, May 26 2005
a(n) is the smallest k such that row k in triangle A265705 contains n. - Reinhard Zumkeller, Dec 17 2015
a(n) is the sum of totient function over powers of 2 <= n. - Anthony Browne, Jun 17 2016
Given positive n, reverse the bits of n and divide by 2^floor(log_2 n). Numerators are in A030101. Ignoring the initial 0, denominators are in this sequence. - Alonso del Arte, Feb 11 2020

Links

Crossrefs

See A000035 for least significant bit(n).
MASKTRANS transform of A055975 (prepended with 0), MASKTRANSi transform of A048678.
Bisection of A065267, A065279, A065291, A072376.
First differences of A063915. Cf. A076877, A073121.
This is Guy Steele's sequence GS(5, 5) (see A135416).
Equals for n >= 1 the first right hand column of A160464. - Johannes W. Meijer, May 24 2009
Diagonal of A088370. - Alois P. Heinz, Oct 28 2011
Cf. A265705, A000010.

Programs

  • Haskell
    a053644 n = if n <= 1 then n else 2 * a053644 (div n 2)
-- Reinhard Zumkeller, Aug 28 2014
a053644_list = 0 : concat (iterate (\zs -> map (* 2) (zs ++ zs)) [1])
-- Reinhard Zumkeller, Dec 08 2012, Oct 21 2011, Oct 17 2010
  • Magma
    [0] cat [2^Ilog2(n): n in [1..90]]; // Vincenzo Librandi, Dec 11 2018
  • Maple
    a:= n-> 2^ilog2(n):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 20 2016
  • Mathematica
    A053644[n_] := 2^(Length[ IntegerDigits[n, 2]] - 1); A053644[0] = 0; Table[A053644[n], {n, 0, 74}] (* Jean-François Alcover, Dec 01 2011 *)
nv[n_] := Module[{c = 2^n}, Table[c, {c}]]; Join[{0}, Flatten[Array[nv, 7, 0]]] (* Harvey P. Dale, Jul 17 2012 *)
  • PARI
    a(n)=my(k=1);while(k<=n,k<<=1);k>>1 \\ Charles R Greathouse IV, May 27 2011
  • PARI
    a(n) = if(!n, 0, 2^exponent(n)) \\ Iain Fox, Dec 10 2018
  • Python
    def a(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1) # Indranil Ghosh, May 25 2017
  • Python
    def A053644(n): return 1<Chai Wah Wu, Jul 27 2022
  • Scala
    (0 to 127).map(Integer.highestOneBit()) // _Alonso del Arte, Feb 26 2020

Formula

a(n) = a(floor(n / 2)) * 2.
a(n) = 2^A000523(n).
From n >= 1 onward, A053644(n) = A062383(n)/2.
a(0) = 0, a(1) = 1 and a(n+1) = a(n)*floor(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1 - x) * (x + Sum_{k >= 1} 2^(k - 1)*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = (A003817(n) + 1)/2 = A091940(n) + 1. - Reinhard Zumkeller, Feb 15 2004
a(n) = Sum_{k = 1..n} (floor(2^k/k) - floor((2^k - 1)/k))*A000010(k). - Anthony Browne, Jun 17 2016
a(2^m+k) = 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016

A065278 A065276 conjugated with A059893, inverse of A065277.

Original entry on oeis.org

2, 4, 1, 8, 3, 10, 6, 16, 5, 18, 7, 20, 12, 22, 14, 32, 9, 34, 11, 36, 13, 38, 15, 40, 24, 42, 26, 44, 28, 46, 30, 64, 17, 66, 19, 68, 21, 70, 23, 72, 25, 74, 27, 76, 29, 78, 31, 80, 48, 82, 50, 84, 52, 86, 54, 88, 56, 90, 58, 92, 60, 94, 62, 128, 33, 130, 35, 132, 37, 134, 39
Offset: 1

Views

Author

Antti Karttunen, Oct 28 2001

Keywords

Comments

This permutation of N induces also such permutation of Z, that p(i)-i >= 0 for all i.

Links

Crossrefs

The associated siteswap sequence is A065279.

Formula

a(n) = A059893(A065276(A059893(n)))
Showing 1-3 of 3 results.

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