A006519 -id:A006519 - cp's OEIS frontend

A283980 a(n) = A003961(n)*A006519(n).

1, 6, 5, 36, 7, 30, 11, 216, 25, 42, 13, 180, 17, 66, 35, 1296, 19, 150, 23, 252, 55, 78, 29, 1080, 49, 102, 125, 396, 31, 210, 37, 7776, 65, 114, 77, 900, 41, 138, 85, 1512, 43, 330, 47, 468, 175, 174, 53, 6480, 121, 294, 95, 612, 59, 750, 91, 2376, 115, 186, 61, 1260, 67, 222, 275, 46656, 119, 390, 71, 684, 145, 462, 73, 5400, 79, 246, 245

Offset: 1

Antti Karttunen, Mar 19 2017

Completely multiplicative since both A003961 and A006519 are. - Andrew Howroyd, Jul 25 2018

			From _Michael De Vlieger_, Dec 29 2019: (Start)
a(1) = 1 since 1 is the empty product.
a(2) = 6 because 2 = 2^1 in form p_k^e; switching p_(k+1) for p, we have 3^1 = 3, and the largest power of 2 dividing 2 is 2^1 = 2; thus 3 * 2 = 6.
a(4) = 36 since 4 = 2^2 -> 4(3^2).
a(6) = 30 since 6 = 2^1 * 3^1 -> 2(3 * 5).
a(12) = 180 since 12 = 2^2 * 3 -> 4(3^2 * 5) = 4(45) = 180.
a(30) = 210 since 30 = 2 * 3 * 5 -> 2(3 * 5 * 7) = 210.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A003961, A006519, A283477.

Array[(Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2] &, 75] (* Michael De Vlieger, Dec 29 2019 *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, 6, nextprime(p+1))^e)} \\ Andrew Howroyd, Jul 25 2018

from sympy import nextprime, factorint
from math import prod
def A283980(n): return prod(nextprime(p)**e if p > 2 else 6**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 08 2022

(define (A283980 n) (* (A006519 n) (A003961 n)))

a(n) = A003961(n)*A006519(n).
From Michael De Vlieger, Dec 29 2019: (Start)
a(p_k) = p_(k+1) for odd prime p.
a(2^k) = 6^k.
a(p_k#) = p_(k+1)# for p_k# = A002110(k). (End)

A100338 Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).

Original entry on oeis.org

1, 3, 5, 3, 8, 7, 1, 1, 2, 8, 4, 2, 9, 8, 8, 2, 3, 7, 4, 3, 8, 8, 8, 9, 4, 0, 8, 4, 0, 1, 6, 6, 0, 8, 1, 2, 4, 2, 2, 7, 3, 3, 3, 4, 1, 6, 8, 1, 2, 1, 1, 8, 5, 5, 6, 9, 2, 3, 6, 7, 2, 6, 4, 9, 7, 8, 7, 0, 0, 1, 8, 4, 2, 0, 6, 4, 8, 2, 6, 0, 5, 4, 8, 4, 3, 1, 9, 6, 9, 7, 6, 0, 1, 7, 4, 6, 5, 6, 9, 7, 9, 6, 6, 8, 5

Offset: 1

Paul D. Hanna, Nov 17 2004

This constant x has the special property that the continued fraction expansion of 2*x results in the continued fraction expansion of x interleaved with 2's: contfrac(x) = [1;2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,...A006519(n),... ] while contfrac(2*x) = [2;1, 2,2, 2,1, 2,4, 2,1, 2,2, 2,1, 2,8,... 2, A006519(n),...].

The continued fraction of x^2 has large partial quotients (see A100864, A100865) that appear to be doubly exponential.

			1.353871128429882374388894084016608124227333416812118556923672649787001842...

Dzmitry Badziahin and Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015.
Dzmitry Badziahin and Jeffrey Shallit, An unusual continued fraction, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.

Cf. A006519, A100863, A100864, A100865.

cf = ContinuedFraction[ Table[ 2^IntegerExponent[n, 2], {n, 1, 200}]]; RealDigits[ FromContinuedFraction[cf // Flatten] , 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)

/* This PARI code generates 1000 digits of x very quickly: */ {x=sqrt(2);y=x;L=2^10;for(i=1,10,v=contfrac(x,2*L); if(2*L>#v,v=concat(v,vector(2*L-#v+1,j,1))); if(2*L>#w,w=concat(w,vector(2*L-#w+1,j,1))); w=vector(2*L,n,if(n%2==1,2,w[n]=v[n\2]));w[1]=floor(2*x); CFW=contfracpnqn(w);x=CFW[1,1]/CFW[2,1]*1.0/2;);x}

{CFM=contfracpnqn(vector(1500,n,2^valuation(n,2))); x=CFM[1,1]/CFM[2,1]*1.0}

A159885 For n >= 1, let f(2n+1) = (3n+2)/A006519(3n+2) and let f^k be the k-th iteration of f. Then a(n) is the least k such that A000120(f^k(2n+1)) <= A000120(n).

Original entry on oeis.org

2, 1, 2, 6, 1, 1, 2, 3, 3, 1, 1, 4, 1, 1, 2, 8, 2, 3, 3, 39, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 2, 8, 5, 2, 2, 41, 3, 2, 3, 5, 5, 1, 1, 1, 1, 1, 1, 42, 2, 1, 4, 6, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 44, 5, 5, 5, 31, 5, 2, 2, 41, 7, 1, 3, 3, 3, 2, 3, 34, 3, 5, 13, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42, 8, 1, 2, 4, 1

Offset: 1

Vladimir Shevelev, Apr 25 2009, Apr 27 2009

Conjecture: a(n) exists for every n >= 1. It is easy to see that this conjecture is equivalent to the well-known Collatz 3x+1 conjecture.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006519, A007814, A000120, A159945, A160198, A160266.

A006519(n) = (1<A006519((3*((n-1)/2))+2); \\ Defined only for odd n. Cf. A075677.
A159885(n) = { my(w=hammingweight(n), n = (n+n+1)); for(k=1,oo,n = f(n); if(hammingweight(n) <= w, return(k))); }; \\ Antti Karttunen, Sep 22 2018

Edited by N. J. A. Sloane, May 03 2009

a(25) corrected, sequence extended by R. J. Mathar, May 15 2009

A006520 Partial sums of A006519.

Original entry on oeis.org

1, 3, 4, 8, 9, 11, 12, 20, 21, 23, 24, 28, 29, 31, 32, 48, 49, 51, 52, 56, 57, 59, 60, 68, 69, 71, 72, 76, 77, 79, 80, 112, 113, 115, 116, 120, 121, 123, 124, 132, 133, 135, 136, 140, 141, 143, 144, 160, 161, 163, 164, 168, 169, 171, 172, 180, 181, 183, 184, 188, 189

Offset: 1

N. J. A. Sloane, Simon Plouffe

The subsequence of primes in this partial sum begins: 3, 11, 23, 29, 31, 59, 71, 79, 113, 163, 181. The subsequence of powers in this partial sum begins: 1, 4, 8, 9, 32, 49, 121, 144, 169. - Jonathan Vos Post, Feb 18 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1001 [Jul 23 2013; offset adapted by _Georg Fischer_, Jan 27 2020]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 44.
Victor Meally, Letter to N. J. A. Sloane, May 1975.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions.
Ralf Stephan, Table of generating functions.

First differences of A022560.

Cf. A001620, A001787, A006519, A136013, A159699.

Table[ 2^IntegerExponent[n, 2], {n, 1, 70}] // Accumulate (* Jean-François Alcover, May 14 2013 *)

PARI
```
a(n)=sum(i=1,n,2^valuation(i,2))
```

def A006520(n): return sum(i&-i for i in range(1,n+1)) # Chai Wah Wu, Jul 14 2022

a(n)/(n*log(n)) is bounded. - Benoit Cloitre, Dec 17 2002
G.f.: (1/(x*(1-x))) * (x/(1-x) + Sum_{k>=1} 2^(k-1)*x^2^k/(1-x^2^k)). - Ralf Stephan, Apr 17 2003
a(n) = b(n+1), with b(2n) = 2b(n) + n, b(2n+1) = 2b(n) + n + 1. - Ralf Stephan, Sep 07 2003
a(2^k-1) = k*2^(k-1) = A001787(k) for any k > 0. - Rémy Sigrist, Jan 21 2021
a(n) ~ (1/(2*log(2)))*n*log(n) + (3/4 + (gamma-1)/(2*log(2)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
a(n) = A136013(n) + n = A159699(n) - n. - Alan Michael Gómez Calderón, Apr 13 2025

More terms from Benoit Cloitre, Dec 17 2002

Offset changed to 1 by N. J. A. Sloane, Oct 18 2019

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 20 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.

The equivalent sequence for integer partitions is A206437.

			---------------------------------------------------------
.              Diagram                Triangle
Compositions     of            of compositions (rows)
.   of 5       regions          and regions (columns)
----------------------------------------------------------
.             _ _ _ _ _
.         5  |_        |                                5
.       1+4  |_|_      |                              1 4
.       2+3  |_  |     |                            2   3
.     1+1+3  |_|_|_    |                          1 1   3
.       3+2  |_    |   |                        3       2
.     1+2+2  |_|_  |   |                      1 2       2
.     2+1+2  |_  | |   |                    2   1       2
.   1+1+1+2  |_|_|_|_  |                  1 1   1       2
.       4+1  |_      | |                4               1
.     1+3+1  |_|_    | |              1 3               1
.     2+2+1  |_  |   | |            2   2               1
.   1+1+2+1  |_|_|_  | |          1 1   2               1
.     3+1+1  |_    | | |        3       1               1
.   1+2+1+1  |_|_  | | |      1 2       1               1
.   2+1+1+1  |_  | | | |    2   1       1               1
. 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
.             _ _ _ _
.         4  |_      |                  4
.       1+3  |_|_    |                1   3
.       2+2  |_  |   |              2       2
.     1+1+2  |_|_|_  |            1   1       2
.       3+1  |_    | |          3               1
.     1+2+1  |_|_  | |        1   2               1
.     2+1+1  |_  | | |      2       1               1
.   1+1+1+1  |_|_|_|_|    1   1       1               1
.
Illustration of the four sections of the set of compositions of 4:
.                                      _ _ _ _
.                                     |_      |     4
.                                     |_|_    |   1+3
.                                     |_  |   |   2+2
.                       _ _ _         |_|_|_  | 1+1+2
.                      |_    |   3          | |     1
.             _ _      |_|_  | 1+2          | |     1
.     _      |_  | 2       | |   1          | |     1
.    |_| 1     |_| 1       |_|   1          |_|     1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j:  1      2    3        4     5      6    7          8
k
--------------------------------------------------------
.  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
2        |_|1        |_  |2         |_|1        |_    |3
3                      | |1                       |   |2
4                      |_|1                       |_  |2
5                                                   | |1
6                                                   | |1
7                                                   | |1
8                                                   |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];

Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.

A228371 First differences of A228370. Also A001511 and A006519 interleaved.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 5, 16, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 6, 32, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 5, 16, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 7, 64

Offset: 1

Omar E. Pol, Aug 21 2013

Number of toothpicks added at n-th stage to the toothpick structure (related to integer compositions) of A228370.

The equivalent sequence for integer partitions is A220517.

			Illustration of the structure after 32 stages. The diagram represents the 16 compositions of 5. The k-th horizontal line segment has length A001511(k) equals the largest part of the k-th region. The k-th vertical line segment has length A006519(k) equals the number of parts of the k-th region.
.      _ _ _ _ _
16     _        |
15     _|_      |
14     _  |     |
13     _|_|_    |
12     _    |   |
11     _|_  |   |
10     _  | |   |
9      _|_|_|_  |
8      _      | |
7      _|_    | |
6      _  |   | |
5      _|_|_  | |
4      _    | | |
3      _|_  | | |
2      _  | | | |
1       | | | | |
.
Written as an irregular triangle the sequence begins:
  1,1;
  2,2;
  1,1,3,4;
  1,1,2,2,1,1,4,8;
  1,1,2,2,1,1,3,4,1,1,2,2,1,1,5,16;
  1,1,2,2,1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,6,32;
  ...

Row lengths give 2*A011782. Right border gives A000079.

Cf. A001511, A006519, A139250, A139251, A206437, A220517, A228370.

def A228371(n): return ((m:=(n>>1)+1)&-m).bit_length() if n&1 else (m:=n>>1)&-m # Chai Wah Wu, Jul 14 2022

a(2n-1) = A001511(n), n >= 1. a(2n) = A006519(n), n >= 1.

A084458 An infinite juggling sequence using three balls: a(n) is the least (lowest) throw needed to ensure that none of the juggling states A084457[n..n+A006519(n)-1] occurs in A084457[0..n-1].

Original entry on oeis.org

4, 4, 6, 0, 5, 8, 0, 0, 3, 7, 0, 7, 0, 9, 0, 0, 7, 0, 10, 0, 0, 0, 7, 12, 0, 0, 0, 0, 4, 10, 0, 0, 9, 0, 0, 11, 0, 0, 0, 9, 0, 13, 0, 0, 0, 0, 9, 0, 14, 0, 0, 0, 0, 0, 5, 12, 0, 0, 0, 11, 0, 0, 14, 0, 0, 0, 0, 11, 0, 0, 15, 0, 0, 0, 0, 0, 10, 0, 16, 0, 0, 0, 0, 0, 0, 5, 13, 0, 0, 0, 13, 0, 0, 0, 15, 0, 0, 0

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

The condition is equivalent to that A000265(A084457(n)) does not occur in A084457[0..n-1].

Index entries for sequences related to juggling
A. Karttunen, Scheme-program for computing this sequence

Variant: A084452. Positions of zeros, minus one: A084463.

A100661 Quet transform of A006519 (see A101387 for definition). Also, least k such that n+k has at most k ones in its binary representation.

Original entry on oeis.org

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

Offset: 1

David Wasserman, Jan 14 2005

If n+a(n) has exactly a(n) 1's in binary, then a(n+1) = a(n)+1, but if n+a(n) has less than a(n) 1's, then a(n+1) = a(n)-1. a(n) is the number of terms needed to represent n as a sum of numbers of the form 2^k-1. [Jeffrey Shallit]
Is a(n) = A080468(n+1)+1?
Compute a(n) by repeatedly subtracting the largest number 2^k-1<=n until zero is reached. The number of times a term was subtracted gives a(n). Examples: 5 = 3 + 1 + 1 ==> a(5) = 3 6 = 3 + 3 ==> a(6) = 2. Replace all zeros in A079559 by -1, then the a(n) are obtained as cumulative sums (equivalent to the generating function given); see fxtbook link. [Joerg Arndt, Jun 12 2006]

			a(4) = 2 because 4+2 (110) has two 1's, but 4+1 (101) has more than one 1.
Conjecture (Joerg Arndt):
a(n) is the number of bits in the binary words of sequence A108918
......A108918.A108918..n..=..n.=.(sum.of.term.2^k-1)
........00001.1.....00001.=..1.=..1
........00011.2.....00010.=..2.=..1.+.1
........00010.1.....00011.=..3.=..3
........00101.2.....00100.=..4.=..3.+.1
........00111.3.....00101.=..5.=..3.+.1.+.1
........00110.2.....00110.=..6.=..3.+.3
........00100.1.....00111.=..7.=..7
........01001.2.....01000.=..8.=..7.+.1
........01011.3.....01001.=..9.=..7.+.1.+.1
........01010.2.....01010.=.10.=..7.+.3
........01101.3.....01011.=.11.=..7.+.3.+.1
........01111.4.....01100.=.12.=..7.+.3.+.1.+.1
........01110.3.....01101.=.13.=..7.+.3.+.3
........01100.2.....01110.=.14.=..7.+.7
........01000.1.....01111.=.15.=.15

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Joerg Arndt, Matters Computational (The Fxtbook), section 1.26.3, p. 72.
David Wasserman, Quet transform + PARI code [Cached copy]

Cf. A006519, A080468, A101387, A100808.

Records at indices in (essentially) A000325.

hb:= n-> `if`(n=1, 0, 1+hb(iquo (n, 2))):
a:= proc(n) local m, t;
      m:= n;
      for t from 0 while m>0 do
        m:= m - (2^(hb(m+1))-1)
      od; t
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 22 2011

hb[n_] := If[n==1, 0, 1+hb[Quotient[n, 2]]];
a[n_] := Module[{m=n, t}, For[t=0, m>0, t++, m = m - (2^(hb[m+1])-1)]; t];
Array[a, 100] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)

A100661(n)=
{ /* method: repeatedly subtract Mersenne numbers */
    local(m, ct);
    if ( n<=1, return(n) );
    m = 1;
    while ( n>m, m<<=1 );
    m -= 1;
    while ( m>n, m>>=1 );
    /* here m=2^k-1 and m<=n */
    ct = 0;
    while ( n, while (m<=n, n-=m; ct+=1);  m>>=1 );
    return( ct );
}
vector(100,n,A100661(n)) /* show terms */
/* Joerg Arndt, Jan 22 2011 */

TInverse(v)=
{
    local(l, w, used, start, x);
    l = length(v); w = vector(l); used = vector(l); start = 1;
    for (i = 1, l,
        while (start <= l && used[start], start++);
        x = start;
        for (j = 2, v[i], x++; while (x <= l && used[x], x++));
        if (x > l,
            return (vector(i - 1, k, w[k]))
            , /* else */
            w[i] = x; used[x] = 1
        )
    );
    return(w);
}
PInverse(v)=
{
    local(l, w);
    l = length(v); w = vector(l);
    for (i = 1, l, if (v[i] <= l, w[v[i]] = i));
    return(w);
}
T(v)=
{
    local(l, w, c);
    l = length(v); w = vector(l);
    for (n = 1, l,
        if (v[n],
            c = 0;
            for (m = 1, n - 1, if (v[m] < v[n], c++));
            w[n] = v[n] - c
            , /* else */
            return (vector(n - 1, i, w[i]))
        )
    );
    return(w);
}
Q(v)=T(PInverse(TInverse(v)));
/* compute terms: */
v = vector(150);
for (n = 1, 150, m = n; x = 1; while (!(m%2), m\=2; x *= 2); v[n] = x); Q(v)

A100661 = lambda n: next(k for k in PositiveIntegers() if (n+k).digits(base=2).count(1) <= k) # D. S. McNeil, Jan 23 2011

a(2^k-1) = 1. For 2^k <= n <= 2^(k+1)-2, a(n) = a(n-2^k+1)+1.

G.f.: x*(2*(1-x)*prod(n>=1, (1+x^(2^n-1))) - 1)/((1-x)^2) = x*(1 + 2*x + 1*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 1*x^6 + 2*x^7 + 3*x^8 + 2*x^9 + ...) [Joerg Arndt, Jun 12 2006]

A101119 Nonzero differences of A006519 (highest power of 2 dividing n) and A003484 (Radon function).

Original entry on oeis.org

7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 239, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 494, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 239, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 1004, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 239

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

nonn

A006519 and A003484 differ only at every 16th term; this sequence forms the nonzero differences. Records form A101120. Equals the XOR BINOMIAL transform of A101122.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003484, A006519, A101120, A101122.

[2^Valuation(16*n,2) - 8*Floor(Valuation(16*n,2)/4) - 2^(Valuation(16*n,2) mod 4): n in [1..50]]; // G. C. Greubel, Nov 01 2018

Table[2^(IntegerExponent[16*n, 2]) - 8*Floor[IntegerExponent[16*n, 2]/4] - 2^(Mod[IntegerExponent[16*n, 2], 4]), {n, 1, 50}] (* G. C. Greubel, Nov 01 2018 *)

{a(n)=2^valuation(16*n,2)-(8*(valuation(16*n,2)\4)+2^(valuation(16*n,2)%4))}

def A101119(n): return (1<<(m:=(~n&n-1).bit_length()+4))-((m&-4)<<1)-(1<<(m&3)) # Chai Wah Wu, Jul 10 2022

a(n) = A006519(16*n) - A003484(16*n) for n>=1. a(2*n-1) = 7 for n>=1.

A118821 2-adic continued fraction of zero, where a(n) = 2 if n is odd, -A006519(n/2) otherwise.

Original entry on oeis.org

2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -8, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -16, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -8, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -32, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -8, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -16

Offset: 1

Paul D. Hanna, May 01 2006

Limit of convergents equals zero; only the 6th convergent is indeterminate. Other 2-adic continued fractions of zero are A118824, A118827, A118830. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.

			For n >= 1, convergents A118822(k)/A118823(k) are:
  at k = 4*n: -1/A080277(n);
  at k = 4*n+1: -2/(2*A080277(n)-1);
  at k = 4*n+2: -1/(A080277(n)-1);
  at k = 4*n-1: 0/(-1)^n.
Convergents begin:
  2/1, -1/-1, 0/-1, -1/1, -2/1, 1/0, 0/1, 1/-4,
  2/-7, -1/3, 0/-1, -1/5, -2/9, 1/-4, 0/1, 1/-12,
  2/-23, -1/11, 0/-1, -1/13, -2/25, 1/-12, 0/1, 1/-16,
  2/-31, -1/15, 0/-1, -1/17, -2/33, 1/-16, 0/1, 1/-32, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006519, A080277; convergents: A118822/A118823; variants: A118824, A118827, A118830; A100338.

Array[-2^(IntegerExponent[#, 2] - 1) /. -1/2 -> 2 &, 96] (* Michael De Vlieger, Nov 02 2018 *)

a(n)=local(p=+2,q=-1);if(n%2==1,p,q*2^valuation(n/2,2))

cp's OEIS Frontend

A283980 a(n) = A003961(n)*A006519(n).

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A100338 Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).

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A159885 For n >= 1, let f(2n+1) = (3n+2)/A006519(3n+2) and let f^k be the k-th iteration of f. Then a(n) is the least k such that A000120(f^k(2n+1)) <= A000120(n).

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A006520 Partial sums of A006519.

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A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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A228371 First differences of A228370. Also A001511 and A006519 interleaved.

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A084458 An infinite juggling sequence using three balls: a(n) is the least (lowest) throw needed to ensure that none of the juggling states A084457[n..n+A006519(n)-1] occurs in A084457[0..n-1].

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A100661 Quet transform of A006519 (see A101387 for definition). Also, least k such that n+k has at most k ones in its binary representation.

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