A214489 -id:A214489 - cp's OEIS frontend

A070939 Length of binary representation of n.

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

N. J. A. Sloane, May 18 2002

Zero is assumed to be represented as 0.
For n>1, n appears 2^(n-1) times. - Lekraj Beedassy, Apr 12 2006
a(n) is the permanent of the n X n 0-1 matrix whose (i,j) entry is 1 iff i=1 or i=j or i=2*j. For example, a(4)=3 is per([[1, 1, 1, 1], [1, 1, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]). - David Callan, Jun 07 2006
a(n) is the number of different contiguous palindromic bit patterns in the binary representation of n; for examples, for 5=101_2 the bit patterns are 0, 1, 101; for 7=111_2 the corresponding patterns are 1, 11, 111; for 13=1101_2 the patterns are 0, 1, 11, 101. - Hieronymus Fischer, Mar 13 2012
A103586(n) = a(n + a(n)); a(A214489(n)) = A103586(A214489(n)). - Reinhard Zumkeller, Jul 21 2012
Number of divisors of 2^n that are <= n. - Clark Kimberling, Apr 21 2019

			8 = 1000 in binary has length 4.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
L. Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.

T. D. Noe, Table of n, a(n) for n = 0..1024
K. Hessami Pilehrood, and T. Hessami Pilehrood, Vacca-Type Series for Values of the Generalized Euler Constant Function and its Derivative, J. Integer Sequences, 13 (2010), #10.7.3.
L. Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for "core" sequences
Index entries for sequences related to binary expansion of n

Cf. A070940, A070941, A001511, A000523.
A029837(n+1) gives the length of binary representation of n without the leading zeros (i.e., when zero is represented as the empty sequence). For n > 0 this is equal to a(n).
This is Guy Steele's sequence GS(4, 4) (see A135416).
Cf. A000120, A007088, A023416, A059015, A113473.
Cf. A083652 (partial sums).

a070939 n = if n < 2 then 1 else a070939 (n `div` 2) + 1
a070939_list = 1 : 1 : l [1] where
   l bs = bs' ++ l bs' where bs' = map (+ 1) (bs ++ bs)
-- Reinhard Zumkeller, Jul 19 2012, Jun 07 2011

A070939:=func< n | n eq 0 select 1 else #Intseq(n, 2) >; [ A070939(n): n in [0..104] ]; // Klaus Brockhaus, Jan 13 2011

A070939 := n -> `if`(n=0, 1, ilog2(2*n)):
seq(A070939(n), n=0..104); # revised by Peter Luschny, Aug 10 2017

Table[Length[IntegerDigits[n, 2]], {n, 0, 50}] (* Stefan Steinerberger, Apr 01 2006 *)
Join[{1},IntegerLength[Range[110],2]] (* Harvey P. Dale, Aug 18 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], BitLength[n]]; (* Michael Somos, Jul 10 2018 *)

{a(n) = if( n<1, n==0, #binary(n))} /* Michael Somos, Aug 31 2012 */

apply( {A070939(n)=exponent(n+!n)+1}, [0..99]) \\ works for negative n and is much faster than the above. - M. F. Hasler, Jan 04 2014, updated Feb 29 2020

def a(n): return len(bin(n)[2:])
print([a(n) for n in range(105)]) # Michael S. Branicky, Jan 01 2021

def A070939(n): return 1 if n == 0 else n.bit_length() # Chai Wah Wu, May 12 2022

def A070939(n) : return (2*n).exact_log(2) if n != 0 else 1
[A070939(n) for n in range(100)] # Peter Luschny, Aug 08 2012

a(0) = 1; for n >= 1, a(n) = 1 + floor(log_2(n)) = 1 + A000523(n).
G.f.: 1 + 1/(1-x) * Sum(k>=0, x^2^k). - Ralf Stephan, Apr 12 2002
a(0)=1, a(1)=1 and a(n) = 1+a(floor(n/2)). - Benoit Cloitre, Dec 02 2003
a(n) = A000120(n) + A023416(n). - Lekraj Beedassy, Apr 12 2006
a(2^m + k) = m + 1, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 14 2017
a(n) = A113473(n) if n>0.

a(4) corrected by Antti Karttunen, Feb 28 2003

A105025 Write numbers in binary under each other; to get the next block of 2^k (k >= 0) terms of the sequence, start at 2^k, read diagonals in downward direction and convert to decimal.

Original entry on oeis.org

0, 1, 3, 2, 4, 7, 6, 5, 11, 10, 9, 12, 15, 14, 13, 8, 18, 17, 20, 23, 22, 21, 16, 27, 26, 25, 28, 31, 30, 29, 24, 19, 33, 36, 39, 38, 37, 32, 43, 42, 41, 44, 47, 46, 45, 40, 35, 50, 49, 52, 55, 54, 53, 48, 59, 58, 57, 60, 63, 62, 61, 56, 51, 34, 68, 71, 70, 69, 64, 75, 74, 73, 76

Offset: 0

N. J. A. Sloane, Apr 03 2005

This is a permutation of the nonnegative integers.

a(A214433(n)) = A105027(A214433(n)); a(A214489(n)) = A105029(A214489(n)). - Reinhard Zumkeller, Jul 21 2012

			........0
........1
.......10
.......11
......100 <- Starting here, the downward diagonals
......101 read 100, 111, 110, 101, giving the block 4, 7, 6, 5.
......110
......111
.....1000
.....1001
.....1010
.....1011
.........

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

Cf. A102370, A105026.
Cf. A070939, A000079.
Cf. A105271 (fixed points), A214416 (inverse).

import Data.Bits ((.|.), (.&.))
a105025 n = foldl (.|.) 0 $ zipWith (.&.)
                  a000079_list $ reverse $ enumFromTo n (n - 1 + a070939 n)
-- Reinhard Zumkeller, Jul 21 2012

a:=proc(i,j) if j=1 and i<=16 then 0 else convert(i+15,base,2)[7-j] fi end: seq(a(i,2)*2^4+a(i+1,3)*2^3+a(i+2,4)*2^2+a(i+3,5)*2+a(i+4,6),i=1..16); # this is a Maple program (not necessarily the simplest) only for one block of (2^4) numbers # Emeric Deutsch, Apr 16 2005

numberOfBlocks = 7; bloc[n_] := Join[ Table[ IntegerDigits[k, 2], {k, 2^(n-1), 2^n-1}], Table[ Rest @ IntegerDigits[k, 2], {k, 2^n, 2^n+n}]]; Join[{0, 1}, Flatten[ Table[ Table[ Diagonal[bloc[n], k] // FromDigits[#, 2]&, {k, 0, -2^(n-1)+1, -1}], {n, 2, numberOfBlocks}]]] (* Jean-François Alcover, Nov 03 2016 *)

More terms from Emeric Deutsch, Apr 16 2005

A103586 a(0)=1, for n > 0: n-th run consists of 2^n-1 copies of n+1.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Benoit Cloitre, Mar 24 2005

a(A214489(n)) = A070939(A214489(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
O. Kullmann and X. Zhao, Bounds for variables with few occurrences in conjunctive normal forms, arXiv preprint arXiv:1408.0629 [math.CO], 2014-2017.
Ana Luzón, Manuel A. Morón, and Luis Felipe Prieto-Martínez, Commutators and commutator subgroups of the Riordan group, (2021).
Index entries for sequences related to binary expansion of n

Number of bits in binary representation of A102370(n).

Cf. A000225.

a103586 n = a070939 (n + a070939 n)
a103586_list = 1 : concat
   (zipWith (replicate . fromInteger) (tail a000225_list) [2..])
-- Reinhard Zumkeller, Jul 21 2012

Join[{1},Flatten[Table[PadRight[{},2^n-1,n+1],{n,6}]]] (* Harvey P. Dale, Aug 22 2021 *)

def A103586(n): return (m:=n.bit_length())+(n>=(1<Chai Wah Wu, Jun 30 2024

a(n) = A070939(n + A070939(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

a(0) = 1 added, definition and offset adjusted by Reinhard Zumkeller, Jul 21 2012

A105029 Write numbers in binary under each other, left justified, read diagonals in downward direction, convert to decimal.

Original entry on oeis.org

0, 2, 6, 5, 4, 14, 13, 8, 11, 10, 9, 12, 30, 29, 24, 19, 18, 17, 20, 23, 22, 21, 16, 27, 26, 25, 28, 62, 61, 56, 51, 34, 33, 36, 39, 38, 37, 32, 43, 42, 41, 44, 47, 46, 45, 40, 35, 50, 49, 52, 55, 54, 53, 48, 59, 58, 57, 60, 126, 125, 120, 115, 98, 65, 68, 71, 70

Offset: 0

Benoit Cloitre, Apr 03 2005

All terms are distinct, but the numbers 2^m - 1 are missing.
a(n) = Sum_{k>=1} B(n+k-1,k)*2^(A103586(n)-k) where B(n,k) n>=1, k>=1 is the infinite array:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
.......
where n-th row consists of binary expansion of n followed by 0's.
a(n) = A105025(n) iff A070939(n) = A103586(n), cf. A214489. - Reinhard Zumkeller, Jul 21 2012

			0
1
1 0
1 1
1 0 0
1 0 1
1 1 0
1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
and reading the diagonals downwards we get 0, 10, 110, 101, 100, 1110, 1101, etc.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences related to binary expansion of n

Cf. A102370, A105030, A105025, A105026, A105027, A105028, A105033.

Cf. A000079, A070939, A103586.

import Data.Bits ((.|.), (.&.))
a105029 n = foldl (.|.) 0 $ zipWith (.&.) a000079_list $
   map (\x -> (len + 1 - a070939 x) * x)
       (reverse $ enumFromTo n (n - 1 + len))  where len = a103586 n
-- Reinhard Zumkeller, Jul 21 2012

A214433 Where A105025 and A105027 agree.

Original entry on oeis.org

0, 1, 2, 3, 9, 10, 13, 14, 34, 37, 42, 45, 50, 53, 58, 61, 517, 522, 533, 538, 549, 554, 565, 570, 581, 586, 597, 602, 613, 618, 629, 634, 645, 650, 661, 666, 677, 682, 693, 698, 709, 714, 725, 730, 741, 746, 757, 762, 773, 778, 789, 794, 805, 810, 821, 826

Offset: 1

Reinhard Zumkeller, Jul 21 2012

A105025(a(n)) = A105027(a(n)).

			A105025(1018) = A105027(1018) = 1011, => 1018 is a term: a(80) = 1018;
A105025(1019) = 994 and A105027(1019) = 1014, => 1019 not a term;
A105025(524302) = A105027(524302) = 524317, => a(81) = 524302;
A105025(524303) = 524312 and A105027(524303) = 524300, => 524302 not a term.
-

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A214489.

a214433 n = a214433_list !! (n-1)
a214433_list = [x | x <- [0..], a105025 x == a105027 x]

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A070939 Length of binary representation of n.

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A105025 Write numbers in binary under each other; to get the next block of 2^k (k >= 0) terms of the sequence, start at 2^k, read diagonals in downward direction and convert to decimal.

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A103586 a(0)=1, for n > 0: n-th run consists of 2^n-1 copies of n+1.

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A105029 Write numbers in binary under each other, left justified, read diagonals in downward direction, convert to decimal.

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A214433 Where A105025 and A105027 agree.

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