A025480 -id:A025480 - cp's OEIS frontend

A153733 Remove all trailing 1's in the binary representation of n.

0, 0, 2, 0, 4, 2, 6, 0, 8, 4, 10, 2, 12, 6, 14, 0, 16, 8, 18, 4, 20, 10, 22, 2, 24, 12, 26, 6, 28, 14, 30, 0, 32, 16, 34, 8, 36, 18, 38, 4, 40, 20, 42, 10, 44, 22, 46, 2, 48, 24, 50, 12, 52, 26, 54, 6, 56, 28, 58, 14, 60, 30, 62, 0, 64, 32, 66, 16, 68, 34, 70, 8, 72, 36, 74, 18, 76, 38

Offset: 0

Reinhard Zumkeller, Dec 31 2008

a(n) is also the map n -> A065423(n+1) applied A007814(n+1) times. - Federico Provvedi, Dec 14 2021

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A000265, A007814, A007088.
Cf. A163575.
Cf. A065423, A331739.

a153733 n = if b == 0 then n else a153733 n'  where (n', b) = divMod n 2
-- Reinhard Zumkeller, Jul 22 2014

f:= n -> (n+1)/2^padic:-ordp(n+1,2)-1:
map(f, [$0..100]); # Robert Israel, Mar 18 2018

Table[If[EvenQ[n],n,FromDigits[Flatten[Most[Split[IntegerDigits[n,2]]]],2]],{n,0,100}] (* Harvey P. Dale, Feb 15 2014 *)
a[n_] := BitShiftRight[n + 1, IntegerExponent[n+1, 2]] - 1; a[Range[0,100]] (* Federico Provvedi, Dec 21 2021 *)

A153733(n)=(n+=1)>>valuation(n,2)-1 \\ most efficient variant: use this. - M. F. Hasler, Mar 16 2018

{a(n)=while(bittest(n,0),n>>=1);n} \\ for illustration: as long as there's a trailing bit 1, remove it. - M. F. Hasler, Mar 16 2018

a(n)=for(i=0,n,bittest(n,i)||return(n>>i)) \\ scan the trailing 1's, then remove all of them at once. - M. F. Hasler, Mar 16 2018

def a(n):
    while n&1: n >>= 1
    return n
print([a(n) for n in range(100)]) # Michael S. Branicky, Dec 18 2021

def A153733(n): return n>>(~(n+1)&n).bit_length() # Chai Wah Wu, Jul 08 2022

a(n) = n if n is even, a((n-1)/2) if odd.
a(n)/2 = A025480(n).
a(n) = A000265(n+1) - 1. - M. F. Hasler, Mar 16 2018
a(n) = n - A331739(n+1). - Federico Provvedi, Dec 21 2021

A353654 Numbers whose binary expansion has the same number of trailing 0 bits as other 0 bits.

Original entry on oeis.org

1, 3, 7, 10, 15, 22, 26, 31, 36, 46, 54, 58, 63, 76, 84, 94, 100, 110, 118, 122, 127, 136, 156, 172, 180, 190, 204, 212, 222, 228, 238, 246, 250, 255, 280, 296, 316, 328, 348, 364, 372, 382, 392, 412, 428, 436, 446, 460, 468, 478, 484, 494, 502, 506, 511, 528, 568

Offset: 1

Mikhail Kurkov, Jul 15 2022

Numbers k such that A007814(k) = A086784(k).
To reproduce the sequence through itself, use the following rule: if binary 1xyz is a term then so are 11xyz and 10xyz0 (except for 1 alone where 100 is not a term).
The number of terms with bit length k is equal to Fibonacci(k-1) for k > 1.
Conjecture: 2*A247648(n-1) + 1 with rewrite 1 -> 1, 01 -> 0 applied to binary expansion is the same as a(n) without trailing 0 bits in binary.
Odd terms are positive Mersenne numbers (A000225), because there is no 0 in their binary expansion. - Bernard Schott, Oct 12 2022

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A000120, A000523, A007814, A010056, A025480, A030109, A054429, A060142, A072649, A084471, A086784, A091892, A132665, A133512, A200650, A213911, A232559, A280514, A343152, A348366.
Cf. A356385 (first differences).
Subsequences with same number k of trailing 0 bits and other 0 bits: A000225 (k=0), 2*A190620 (k=1), 4*A357773 (k=2), 8*A360573 (k=3).

N:= 10: # for terms <= 2^N
S:= {1};
for d from 1 to N do
  for k from 0 to d/2-1 do
    B:= combinat:-choose([$k+1..d-2],k);
    S:= S union convert(map(proc(t) local s; 2^d - 2^k - add(2^(s),s=t) end proc,B),set);
od od:
sort(convert(S,list)); # Robert Israel, Sep 21 2023

Join[{1}, Select[Range[2, 600], IntegerExponent[#, 2] == Floor[Log2[# - 1]] - DigitCount[# - 1, 2, 1] &]] (* Amiram Eldar, Jul 16 2022 *)

isok(k) = if (k==1, 1, (logint(k-1, 2)-hammingweight(k-1) == valuation(k, 2))); \\ Michel Marcus, Jul 16 2022

from itertools import islice, count
def A353654_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n & n-1).bit_length()) == bin(n>>m)[2:].count('0'),count(max(startvalue,1)))
A353654_list = list(islice(A353654_gen(),30)) # Chai Wah Wu, Oct 14 2022

a(n) = a(n-1) + A356385(n-1) for n > 1 with a(1) = 1.
Conjectured formulas: (Start)
a(n) = 2^g(n-1)*(h(n-1) + 2^A000523(h(n-1))*(2 - g(n-1))) for n > 2 with a(1) = 1, a(2) = 3 where f(n) = n - A130312(n), g(n) = [n > 2*f(n)] and where h(n) = a(f(n) + 1).
a(n) = 1 + 2^r(n-1) + Sum_{k=1..r(n-1)} (1 - g(s(n-1, k)))*2^(r(n-1) - k) for n > 1 with a(1) = 1 where r(n) = A072649(n) and where s(n, k) = f(s(n, k-1)) for n > 0, k > 1 with s(n, 1) = n.
a(n) = 2*(2 + Sum_{k=1..n-2} 2^(A213911(A280514(k)-1) + 1)) - 2^A200650(n) for n > 1 with a(1) = 1.
A025480(a(n)-1) = A348366(A343152(n-1)) for n > 1.
a(A000045(n)) = 2^(n-1) - 1 for n > 1. (End)

A374993 Total cost when the elements of the n-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the transpose updating strategy.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 3, 4, 4, 3, 5, 4, 5, 4, 4, 5, 5, 6, 5, 5, 5, 6, 6, 5, 5, 4, 6, 5, 6, 5, 5, 6, 6, 6, 6, 5, 7, 7, 6, 6, 8, 4, 6, 7, 8, 7, 7, 6, 6, 7, 6, 6, 6, 7, 7, 6, 6, 5, 7, 6, 7, 6, 6, 7, 7, 7, 7, 8, 8, 7, 7, 7, 7, 8, 8, 6, 8, 8, 7, 7, 8, 6, 10, 6, 6, 7

Offset: 0

Pontus von Brömssen, Jul 27 2024

nonn

The cost of a request equals the position of the requested element in the list.

After a request, the requested element is transposed with the immediately preceding element (unless the requested element is already at the front of the list).

			For n=931 (the smallest n for which A374992(n), A374994(n), A374995(n), and a(n) are all distinct), the 931st composition is (1, 1, 2, 4, 1, 1), giving the following development of the list:
   list   | position of requested element
  --------+------------------------------
  1 2 3 4 |         1
  ^       |
  1 2 3 4 |         1
  ^       |
  1 2 3 4 |         2
    ^     |
  2 1 3 4 |         4
        ^ |
  2 1 4 3 |         2
    ^     |
  1 2 4 3 |         1
  ^       |
  ---------------------------------------
          a(931) = 11

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, 1998, pp. 401-403.

John Tyler Rascoe, Table of n, a(n) for n = 0..10000
Ran Bachrach and Ran El-Yaniv, Online list accessing algorithms and their applications: recent empirical evidence, Proceedings of the 8th annual ACM-SIAM symposium on discrete algorithms, SODA ’97, New Orleans, LA, January 5-7, 1997, 53-62.
Wikipedia, Self-organizing list.

Row n=1 of A374996.
Analogous sequences for other updating strategies: A374992, A374994, A374995.
Cf. A000120, A025480, A066099 (compositions in standard order), A333766, A374998.

def comp(n):
    # see A357625
    return
def A374993(n):
    if n<1: return 0
    cost,c = 0,comp(n)
    m = list(range(1,max(c)+1))
    for i in c:
        j = m.index(i)
        cost += j+1
        if j > 0:
            m.insert(j-1,m.pop(j))
    return cost # John Tyler Rascoe, Aug 02 2024

The sum of a(j) over all j such that A000120(j) = k (number of requests) and A333766(j) <= m (upper bound on the requested elements) equals m^k * k * (m+1)/2. This is a consequence of the fact that the first m positions of the list are occupied by the elements 1, ..., m, as long as no element larger than m has been requested so far.

a(n) = a(A025480(n-1)) + A374998(n) for n >= 1.

A374996 Square array read by antidiagonals: T(n,k) is the total cost when the elements of the k-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the move-ahead(n) updating strategy; n, k >= 0.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 3, 2, 2, 1, 0, 3, 3, 2, 2, 1, 0, 3, 4, 3, 2, 2, 1, 0, 3, 3, 4, 3, 2, 2, 1, 0, 4, 3, 3, 4, 3, 2, 2, 1, 0, 4, 4, 3, 3, 4, 3, 2, 2, 1, 0, 4, 4, 4, 3, 3, 4, 3, 2, 2, 1, 0, 4, 3, 5, 4, 3, 3, 4, 3, 2, 2, 1, 0, 4, 5, 3, 5, 4, 3, 3, 4, 3, 2, 2, 1, 0

Offset: 0

Pontus von Brömssen, Jul 27 2024

The cost of a request equals the position of the requested element in the list.

After a request, the requested element is moved n steps closer to the front of the list (or to the front if the element is already less than n steps from the front).

			Array begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  ---+-----------------------------------------------
   0 | 0  1  2  2  3  3  3  3  4  4  4  4  4  4  4  4
   1 | 0  1  2  2  3  4  3  3  4  4  3  5  4  5  4  4
   2 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   3 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   4 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   5 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   6 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   7 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   8 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
   9 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
  10 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
  11 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
  12 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
  13 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
  14 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4
  15 | 0  1  2  2  3  4  3  3  4  5  3  5  4  5  4  4

John Tyler Rascoe, Antidiagonals n = 0..139, flattened
Ran Bachrach and Ran El-Yaniv, Online list accessing algorithms and their applications: recent empirical evidence, Proceedings of the 8th annual ACM-SIAM symposium on discrete algorithms, SODA ’97, New Orleans, LA, January 5-7, 1997, 53-62.
Wikipedia, Self-organizing list.

Cf. A000120, A025480, A029837 (row n=0), A374993 (row n=1), A333766, A374992, A375001.

def comp(n):
    # see A357625
    return
def A374996(n,k):
    if k<1: return 0
    cost,c = 0,comp(k)
    m = list(range(1,max(c)+1))
    for i in c:
        j = m.index(i)
        cost += j+1
        jp = 0
        if j >= n:
            jp += j-n
        m.insert(jp,m.pop(j))
    return cost # John Tyler Rascoe, Aug 02 2024

T(n,k) = A374992(k) if n >= A333766(k)-1.
The sum of T(n,k) over all k such that A000120(k) = j (number of requests) and A333766(k) <= m (upper bound on the requested elements) equals m^j * j * (m+1)/2. This is a consequence of the fact that the first m positions of the list are occupied by the elements 1, ..., m, as long as no element larger than m has been requested so far.
T(n,k) = T(n,A025480(k-1)) + A375001(n,k) for n >= 0 and k >= 1.

A005766 a(n) = cost of minimal multiplication-cost addition chain for n.

Original entry on oeis.org

0, 1, 3, 5, 9, 12, 18, 21, 29, 34, 44, 48, 60, 67, 81, 85, 101, 110, 128, 134, 154, 165, 187, 192, 216, 229, 255, 263, 291, 306, 336, 341, 373, 390, 424, 434, 470, 489, 527, 534, 574, 595, 637, 649, 693, 716, 762, 768, 816, 841, 891, 905, 957, 984, 1038

Offset: 1

N. J. A. Sloane, Jeffrey Shallit, Robert G. Wilson v

nonn

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

T. D. Noe, Table of n, a(n) for n = 1..1000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197, ex. 21.
R. L. Graham, A. C.-C. Yao, F. F. Yao, Addition chains with multiplicative cost Discrete Math. 23 (1978), 115-119.
R. L. Graham et al., Addition chains with multiplicative cost [Cached copy]
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Index to sequences related to the complexity of n

Partial sums of A089265.

a[n_] := Sum[v = IntegerExponent[k, 2]; v + k/2^v - 1, {k, 1, n}];
Array[a, 55] (* Jean-François Alcover, Feb 28 2019 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n^2/4,a((n-1)/2)+(n-1)*(n+3)/4))

a(n)=sum(k=1,n,valuation(k,2)+k/2^valuation(k,2)-1)

a(2n)=a(n)+n^2, a(2n+1)=a(n)+n(n+2). - Ralf Stephan, May 04 2003
G.f.: 1/(1-x) * sum(k>=0, x^2^(k+1)(1+2x^2^k-x^2^(k+1))/(1-x^2^(k+1))^2). - Ralf Stephan, Jul 27 2003
a(n) = sum(k=1, n, A007814(n) + 2*A025480(n-1)). - Ralf Stephan, Oct 30 2003

More terms from Ralf Stephan, May 04 2003

A006014 a(n+1) = (n+1)a(n) + Sum_{k=1..n-1} a(k)a(n-k).

Original entry on oeis.org

1, 2, 7, 32, 178, 1160, 8653, 72704, 679798, 7005632, 78939430, 965988224, 12762344596, 181108102016, 2748049240573, 44405958742016, 761423731533286, 13809530704348160, 264141249701936818, 5314419112717217792, 112201740111374359516, 2480395343617443024896

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 7*x^3 + 32*x^4 + 178*x^5 + 1160*x^6 + 8653*x^7 + 72704*x^8 + ...

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

Michael De Vlieger, Table of n, a(n) for n = 1..449
Jimmy Devillet and Bruno Teheux, Associative, idempotent, symmetric, and order-preserving operations on chains, arXiv:1805.11936 [math.RA], 2018.
E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, Fighting fish. J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017), chapter 4.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

Cf. A007814, A025480, A073017.

Nest[Append[#1, #1[[-1]] (#2 + 1) + Total@ Table[#1[[k]] #1[[#2 - k]], {k, #2 - 1}]] & @@ {#, Length@ #} &, {1}, 17] (* Michael De Vlieger, Aug 22 2018 *)
(* or *)
a[1] = 1; a[n_] := a[n] = n a[n-1] + Sum[a[k] a[n-1-k], {k, n-2}]; Array[a, 18] (* Giovanni Resta, Aug 23 2018 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = k * A[k-1] + sum( j=1, k-2, A[j] * A[k-1-j])); A[n])} /* Michael Somos, Jul 24 2011 */

from sage.all import *
@CachedFunction
def a(n): # a = A006014
    if n<5: return (pow(5,n-1) + 3)//4
    else: return n*a(n-1) + 2*sum(a(k)*a(n-k-1) for k in range(1,(n//2))) + (n%2)*pow(a((n-1)//2),2)
print([a(n) for n in range(1,61)]) # G. C. Greubel, Jan 10 2025

G.f. A(x) satisfies A(x) = x * (1 + A(x) + A(x)^2 + x * A'(x)). - Michael Somos, Jul 24 2011
Conjecture: a(n) = Sum_{k=0..2^(n-1) - 1} b(k) for n > 0 where b(2n+1) = b(n), b(2n) = b(n) + b(n - 2^f(n)) + b(2n - 2^f(n)) + b(A025480(n-1)) for n > 0 with b(0) = b(1) = 1 and where f(n) = A007814(n). - Mikhail Kurkov, Nov 19 2021
a(n) ~ cosh(sqrt(3)*Pi/2) * n! / Pi = A073017 * n!. - Vaclav Kotesovec, Nov 21 2024

A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 8, 1, 16, 4, 32, 2, 64, 8, 128, 1, 256, 16, 512, 4, 1024, 32, 2048, 2, 4096, 64, 8192, 8, 16384, 128, 32768, 1, 65536, 256, 131072, 16, 262144, 512, 524288, 4, 1048576, 1024, 2097152, 32, 4194304, 2048, 8388608, 2, 16777216

Offset: 0

Ralf Stephan, Jun 07 2003

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A045654, A220466.

nmax := 48: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := 2^n od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 11 2013
A082392 := proc(n)
    2^A025480(n) ;
end proc:
seq(A082392(n),n=0..100) ; # R. J. Mathar, Jul 16 2020

a[n_] := 2^(((n+1)/2^IntegerExponent[n+1, 2]+1)/2-1);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 15 2023 *)

for(n=0, 50, l=ceil(log(n+1)/log(2)); t=polcoeff(sum(k=0, l, (x^2^k)/(1-2*x^2^(k+1)))/x + O(x^(n+1)), n); print1(t", ");) ;

a(0) = 1, a(2*n) = 2^n, a(2*n+1) = a(n).
a(n) = 2^A025480(n) = 2^(A003602(n)-1).
a((2*n+1)*2^p-1) = 2^n, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 11 2013

A108202 a(2n) = A007376(n); a(2n+1) = a(n).

Original entry on oeis.org

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

Offset: 0

Juliette Bruyndonckx and Alexandre Wajnberg, Jun 15 2005

A sequence, S, consisting of the natural counting digits (A007376) interleaved with S.
Fractal sequence (Kimberling-type) based upon the counting digits.
Similar to but different from A025480 - N. J. A. Sloane, Nov 26 2020.

First differs from A025480 at a(20).
Even bisection: A007376.
Cf. A003602.

a(n) = A007376(A025480(n)). - Kevin Ryde, Nov 21 2020

More terms from Joshua Zucker, May 18 2006

Name made more explicit by Peter Munn, Nov 20 2020

A131987 Representation of a dense para-sequence.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 1, 6, 3, 7, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 16, 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 31, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10, 42, 21, 43

Offset: 1

Clark Kimberling, Aug 05 2007, Sep 12 2007

A fractal sequence. The para-sequence may be regarded as a sort of "limit" of the concatenated segments. The para-sequence (itself not a sequence) is dense in the sense that every pair of terms i and j are separated by another term (and hence separated by infinitely many terms).
The para-sequence accounts for positions of dyadic rational numbers in the following way: Label 1/2 as 1; label 1/4, 3/4 as 2 and 3; label 1/8, 3/8, 5/8, 7/8 as 4,5,6,7, etc. Then, for example, the ordering 1/8 < 1/4 < 3/8 < 1/2 < 5/8 < 3/4 < 7/8 matches the labels 4,2,5,1,6,3,7, which is the 3rd segment of A131987. The n-th segment consists of labels for rationals having denominators 2, 4, 8, ..., 2^n.
Could be seen as a "fuzzy table" with row lengths 2^n-1. In row n one has the numbers, read from the leftmost to the rightmost, as they appear in a perfect binary tree of 2^n-1 nodes when inserted in "storage order" into the tree, cf. illustration in A101279 and stackexchange link. These rows are obviously permutations of [1..2^n-1], their inverse is given in A269752. - M. F. Hasler, Mar 04 2016
Subsequence of A025480 (omitting all terms=0). - David James Sycamore, Apr 26 2020
The sequence obtained by adding 1 to every term of this sequence is the same as A003602 with all 1's removed. - David James Sycamore, Jul 25 2022

			Start with 1 and isolate it using 2,3, like this: 2,1,3.  Then isolate those using 4,5,6,7, like this: 4,2,5,1,6,3,7.  The next segment, to be concatenated after 4,2,5,1,6,3,7, is 8,4,9,2,10,5,11,1,12,6,13,3,14,7,15.

C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Clark Kimberling, Table of n, a(n) for n = 1..10000
Josef Eschgfäller, Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.
N.A., References to these functions relating to binary trees and binary digit counting?, Stackexchange forum, Feb 28 2016.
Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Cf. A001511, A025480, A070941, A133108, A003602.

m:=p->padic[ordp](2*p,2)-1:podd:=(h,p)->2^h+(p-2)/2:peven:=(h,p)->2^(h-m(p))+(p-2^m(p))/2^(m(p)+1):for i from 0 to 5 do for j from 1 to 2^(i+1)-1 do if j mod 2 =1 then print(podd(i,j)) else print(peven(i,j)) fi od od # Gary Detlefs, Sep 28 2018

Flatten@NestList[Riffle[Range[Length[#] + 1, 2 Length[#] + 1], #] &, {1}, 4] (* Birkas Gyorgy, Mar 11 2011 *)

A131987_row(n,r=[1])={for(k=2,n,r=vector(2^k-1,j,if(bittest(j,0),j\2+2^(k-1),r[j\2])));r}
apply(A131987_row,[1..6]) \\ or concat(...) \\ M. F. Hasler, Mar 04 2016

When viewed as a table, T(h,p), related to the in order traversal of a full binary tree, T(h,p) = 2^h+(p-1)/2, p odd, 2^(h-m(p)) + (p-2^m(p)) / 2^(m(p)+1), where m(p) is the greatest value of n such that p mod 2^n == 0. m(p) = p-adic[ordp](2*p,2)-1. - Gary Detlefs, Sep 28 2018

a((2*n+1)*2^k - k - A070941(n)) = n = A025480((2*n+1)*2^k - 1); (n>=1, k>=0). - David James Sycamore, Apr 26 2020

A181363 1 followed by the primes, interleaved recursively.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 7, 3, 11, 2, 13, 5, 17, 1, 19, 7, 23, 3, 29, 11, 31, 2, 37, 13, 41, 5, 43, 17, 47, 1, 53, 19, 59, 7, 61, 23, 67, 3, 71, 29, 73, 11, 79, 31, 83, 2, 89, 37, 97, 13, 101, 41, 103, 5, 107, 43, 109, 17, 113, 47, 127, 1, 131, 53, 137, 19, 139, 59, 149, 7, 151, 61

Offset: 1

Reinhard Zumkeller, Oct 16 2010

nonn

a(2*n-1) = A008578(n); a(2*n+1) = A000040(n);
a((2*n-1)*2^k) = A008578(n), k >= 0;
a(2^k) = 1, k >= 0; a(2^k - 1) = A051438(k), k > 0.

			Initial values, seen as a binary tree:
................................. 1
................ 1 _______________________________ 2
........ 1 _____________ 3 .............. 2 ______________ 5
... 1 _____ 7 ..... 3 ______ 11 .... 2 ______ 13 .... 5 ______ 17
.. 1__19...7__23...3__29...11__31...2__37...13__41...5__43...17__47

R. Zumkeller, Table of n, a(n) for n = 1..8191

import Data.List (transpose)
a181363 n = a181363_list !! (n-1)
a181363_list = concat $ transpose [a008578_list, a181363_list]
-- Reinhard Zumkeller, Mar 22 2014, Oct 20 2011, Oct 16 2010

a(n) = if even n then a(n/2) else A008578((n+1)/2).

a(n) = A008578(A025480(n-1)+1). - Reinhard Zumkeller, Mar 22 2014

cp's OEIS Frontend

A153733 Remove all trailing 1's in the binary representation of n.

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A353654 Numbers whose binary expansion has the same number of trailing 0 bits as other 0 bits.

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A374993 Total cost when the elements of the n-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the transpose updating strategy.

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A374996 Square array read by antidiagonals: T(n,k) is the total cost when the elements of the k-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the move-ahead(n) updating strategy; n, k >= 0.

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A005766 a(n) = cost of minimal multiplication-cost addition chain for n.

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A006014 a(n+1) = (n+1)*a(n) + Sum_{k=1..n-1} a(k)*a(n-k).

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A006014 a(n+1) = (n+1)a(n) + Sum_{k=1..n-1} a(k)a(n-k).