A005187 -id:A005187 - cp's OEIS frontend

A055938 Integers not generated by b(n) = b(floor(n/2)) + n (complement of A005187).

2, 5, 6, 9, 12, 13, 14, 17, 20, 21, 24, 27, 28, 29, 30, 33, 36, 37, 40, 43, 44, 45, 48, 51, 52, 55, 58, 59, 60, 61, 62, 65, 68, 69, 72, 75, 76, 77, 80, 83, 84, 87, 90, 91, 92, 93, 96, 99, 100, 103, 106, 107, 108, 111, 114, 115, 118, 121, 122, 123, 124, 125, 126, 129

Offset: 1

Alford Arnold, Jul 21 2000

Note that the lengths of the consecutive runs in a(n) form sequence A001511.
Integers that are not a sum of distinct integers of the form 2^k-1. - Vladeta Jovovic, Jan 24 2003
Also n! never ends in this many 0's in base 2 - Carl R. White, Jan 21 2008
A079559(a(n)) = 0. - Reinhard Zumkeller, Mar 18 2009
These numbers are dead-end points when trying to apply the iterated process depicted in A071542 in reverse, i.e. these are positive integers i such that there does not exist k with A000120(i+k)=k. See also comments at A179016. - Antti Karttunen, Oct 26 2012
Conjecture: a(n)=b(n) defined as b(1)=2, for n>1, b(n+1)=b(n)+1 if n is already in the sequence, b(n+1)=b(n)+3 otherwise. If so, then see Cloitre comment in A080578. - Ralf Stephan, Dec 27 2013
Numbers n for which A257265(m) = 0. - Reinhard Zumkeller, May 06 2015. Typo corrected by Antti Karttunen, Aug 08 2015
Numbers which have a 2 in their skew-binary representation (cf. A169683). - Allan C. Wechsler, Feb 28 2025

			Since A005187 begins 0 1 3 4 7 8 10 11 15 16 18 19 22 23 25 26 31... this sequence begins 2 5 6 9 12 13 14 17 20 21

Complement of A005187. Setwise difference of A213713 and A213717.
Row 1 of arrays A257264, A256997 and also of A255557 (when prepended with 1). Equally: column 1 of A256995 and A255555.
Cf. also arrays A254105, A254107 and permutations A233276, A233278.
Left inverses: A234017, A256992.
Cf. A001511, A046699, A079559, A080578, A086343, A227359, A227408, A234016.
Gives positions of zeros in A213714, A213723, A213724, A213731, A257265, positions of ones in A213725-A213727 and A256989, positions of nonzeros in A254110.
Cf. also A010061 (integers that are not a sum of distinct integers of the form 2^k+1).
Analogous sequence for factorial base number system: A219658, for Fibonacci number system: A219638, for base-3: A096346. Cf. also A136767-A136774.
Cf. A257508, A257509, A257126.

a055938 n = a055938_list !! (n-1)
a055938_list = concat $
   zipWith (\u v -> [u+1..v-1]) a005187_list $ tail a005187_list
-- Reinhard Zumkeller, Nov 07 2011

a[0] = 0; a[1] = 1; a[n_Integer] := a[Floor[n/2]] + n; b = {}; Do[ b = Append[b, a[n]], {n, 0, 105}]; c =Table[n, {n, 0, 200}]; Complement[c, b]
(* Second program: *)
t = Table[IntegerExponent[(2n)!, 2], {n, 0, 100}]; Complement[Range[t // Last], t] (* Jean-François Alcover, Nov 15 2016 *)

L=listcreate();for(n=1,1000,for(k=2*n-hammingweight(n)+1,2*n+1-hammingweight(n+1),listput(L,k)));Vec(L) \\ Ralf Stephan, Dec 27 2013

def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
def a043545(n):
    x=bin(n)[2:]
    return int(max(x)) - int(min(x))
def a079559(n): return 1 if n==0 else a043545(n + 1)*a079559(n + 1 - a053644(n + 1))
print([n for n in range(1, 201) if a079559(n)==0]) # Indranil Ghosh, Jun 11 2017, after the comment by Reinhard Zumkeller

;; utilizing COMPLEMENT-macro from Antti Karttunen's IntSeq-library)
(define A055938 (COMPLEMENT 1 A005187))
;; Antti Karttunen, Aug 08 2015

a(n) = A080578(n+1) - 2 = A080468(n+1) + 2*n (conjectured). - Ralf Stephan, Dec 27 2013
From Antti Karttunen, Aug 08 2015: (Start)
Other identities. For all n >= 1:
A234017(a(n)) = n.
A256992(a(n)) = n.
A257126(n) = a(n) - A005187(n).
(End)

More terms from Robert G. Wilson v, Jul 24 2000

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58

Offset: 1

Antti Karttunen, Nov 25 2017

sign

"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* Michael De Vlieger, Nov 26 2017 *)

A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ Antti Karttunen, Dec 16 2024

(define (A294898 n) (- (A005187 n) (A000203 n)))

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Name edited by Antti Karttunen, Dec 16 2024

A056982 a(n) = 4^A005187(n). The denominators of the Landau constants.

Original entry on oeis.org

1, 4, 64, 256, 16384, 65536, 1048576, 4194304, 1073741824, 4294967296, 68719476736, 274877906944, 17592186044416, 70368744177664, 1125899906842624, 4503599627370496, 4611686018427387904, 18446744073709551616, 295147905179352825856, 1180591620717411303424

Offset: 0

Eric W. Weisstein

Also equal to A046161(n)^2.
Let W(n) = Product_{k=1..n} (1- 1/(4*k^2)), the partial Wallis product with lim n -> infinity W(n) = 2/Pi; a(n) = denominator(W(n)). The numerators are in A069955.
Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n >= 1} 4*n^2/(4*n^2-1). Numerators are in A069955.
Denominator of h^(2n) in the Kummer-Gauss series for the perimeter of an ellipse.
Denominators of coefficients in hypergeometric([1/2,-1/2],[1],x). The numerators are given in A038535. hypergeom([1/2,-1/2],[1],e^2) = L/(2*Pi*a) with the perimeter L of an ellipse with major axis a and numerical eccentricity e (Maclaurin 1742). - Wolfdieter Lang, Nov 08 2010
Also denominators of coefficients in hypergeometric([1/2,1/2],[1],x). The numerators are given in A038534. - Wolfdieter Lang, May 29 2016
Also denominators of A277233. - Wolfdieter Lang, Nov 16 2016
A277233(n)/a(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - Peter Luschny, Sep 27 2019

J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 84. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
O. J. Farrell and B. Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.

Charles R Greathouse IV, Table of n, a(n) for n = 0..500
B. Gourevitch, L'univers de Pi
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the Hathi Trust Digital Library.]
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung), Arch. Math. Phys. 21 (1913), 250-255. [Accessible in the USA through the Hathi Trust Digital Library.]
Cristinel Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011), pp. 1011-1018.
G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.
Eric Weisstein's World of Mathematics, Gauss-Kummer Series
Eric Weisstein's World of Mathematics, Ellipse
Index to divisibility sequences

Apart from offset, identical to A110258.
Cf. A005187, A046161, A056981, A069955.
Equals (1/2)*A038533(n), A038534, A277233.
Cf. A001790/A046161.

A056982 := n -> denom(binomial(1/2, n))^2:
seq(A056982(n), n=0..19); # Peter Luschny, Apr 08 2016
# Alternatively:
G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
[seq(coeff(ser,x,n), n=0..19)]: denom(%); # Peter Luschny, Sep 28 2019

Table[Power[4, 2 n - DigitCount[2 n, 2, 1]], {n, 0, 19}] (* Michael De Vlieger, May 30 2016, after Harvey P. Dale at A005187 *)
G[x_] := (2 EllipticK[x])/(Pi (1 - x));
CoefficientList[Series[G[x], {x, 0, 19}], x] // Denominator (* Peter Luschny, Sep 28 2019 *)

a(n)=my(s=n); while(n>>=1, s+=n); 4^s \\ Charles R Greathouse IV, Apr 07 2012

a(n) = (denominator(binomial(1/2, n)))^2. - Peter Luschny, Sep 27 2019

Edited by N. J. A. Sloane, Feb 18 2004, Jun 05 2007

A213714 Inverse function for injection A005187.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 0, 4, 5, 0, 6, 7, 0, 0, 0, 8, 9, 0, 10, 11, 0, 0, 12, 13, 0, 14, 15, 0, 0, 0, 0, 16, 17, 0, 18, 19, 0, 0, 20, 21, 0, 22, 23, 0, 0, 0, 24, 25, 0, 26, 27, 0, 0, 28, 29, 0, 30, 31, 0, 0, 0, 0, 0, 32, 33, 0, 34, 35, 0, 0, 36, 37, 0, 38, 39, 0, 0, 0, 40, 41, 0, 42, 43, 0, 0, 44, 45, 0, 46, 47, 0

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

a(0)=0; thereafter if n occurs as a term of A005187, a(n)=its position in A005187, otherwise zero. This works as an "inverse" function for A005187 in a sense that a(A005187(n)) = n for all n.

a(n)*A234017(n) = 0 for all n.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Can be used when computing A213715, A213723, A213724, A233275, A233277. Cf. A005187, A046699, A079559, A234017, A230414.

import Data.List (genericIndex)
a213714 n = genericIndex a213714_list n
a213714_list = f [0..] a005187_list 0 where
   f (x:xs) ys'@(y:ys) i | x == y    = i : f xs ys (i+1)
                         | otherwise = 0 : f xs ys' i
-- Reinhard Zumkeller, May 01 2015

from sympy import factorial
def a046699(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1))>0: s+=1
    return s
def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
def a043545(n):
    x=bin(n)[2:]
    return int(max(x)) - int(min(x))
def a079559(n): return 1 if n==0 else a043545(n + 1)*a079559(n + 1 - a053644(n + 1))
def a(n): return 0 if n==0 else a079559(n)*(a046699(n + 2) - 1) # Indranil Ghosh, Jun 11 2017

a(0)=0, for n>0, a(n) = A079559(n) * (A046699(n+2)-1) [With A046699's October 2012 starting offset. Incorrect indexing shown in this formula corrected by Antti Karttunen, Dec 18 2013]

A233275 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with even and odd numbers.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 14, 10, 15, 11, 9, 8, 24, 25, 26, 28, 27, 29, 20, 30, 21, 22, 18, 31, 23, 19, 17, 16, 48, 49, 50, 52, 51, 53, 56, 54, 57, 58, 40, 55, 59, 41, 60, 42, 61, 44, 36, 43, 45, 62, 46, 37, 38, 34, 63, 47, 39, 35, 33, 32, 96, 97, 98, 100

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

It seems that for all n, a(A000079(n)) = A003945(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python Program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233276.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233277/A233278, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=1, a(n) = 2*a(A213714(n)-1), else a(n) = 1+(2*a(A234017(n))).

a(n) = A054429(A233277(n)). [Follows from the definitions of these sequences]

A233276 a(0)=0, a(1)=1, after which a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 11, 13, 8, 9, 10, 12, 31, 30, 26, 29, 22, 24, 25, 28, 16, 17, 18, 20, 19, 21, 23, 27, 63, 62, 57, 61, 50, 55, 56, 60, 42, 45, 47, 51, 49, 52, 54, 59, 32, 33, 34, 36, 35, 37, 39, 43, 38, 40, 41, 44, 46, 48, 53, 58, 127, 126, 120

Offset: 0

Antti Karttunen, Dec 18 2013

For all n, a(A000079(n)) = A000225(n+1), i.e. a(2^n) = (2^(n+1))-1.
For n>=1, a(A000225(n)) = A000325(n).
This permutation is obtained by "entangling" even and odd numbers with complementary pair A005187 & A055938, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A005187(n+1) to the parent node containing n, and each child to the right is obtained as A055938(n):
                                     0
                                     |
                  ...................1...................
                 3                                       2
       7......../ \........6                   4......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  15       14         11       13          8       9          10       12
31  30   26  29     22  24   25  28      16 17   18 20      19  21   23  27
etc.
For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to 0 from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.
Permutation A233278 gives the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233275.
Cf. A005187, A054429, A055938, A256991, A256478, A256479.
Cf. also A070939 (the binary width of both n and a(n)).
Related arrays: A255555, A255557.
Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233277/A233278, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).
As a composition of related permutations:
a(n) = A233278(A054429(n)).

Name changed and the illustration of binary tree added by Antti Karttunen, Apr 19 2015

A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 9, 13, 8, 12, 14, 15, 23, 22, 21, 19, 20, 18, 27, 17, 26, 25, 29, 16, 24, 28, 30, 31, 47, 46, 45, 43, 44, 42, 39, 41, 38, 37, 55, 40, 36, 54, 35, 53, 34, 51, 59, 52, 50, 33, 49, 58, 57, 61, 32, 48, 56, 60, 62, 63, 95, 94, 93, 91

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Indranil Ghosh, PARI program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233278.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=0, a(n) = 2*a(A234017(n)), else a(n) = 1+(2*a(A213714(n)-1)).

a(n) = A054429(A233275(n)). [Follows from the definitions of these sequences]

A233278 a(0)=0, a(1)=1, after which a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 12, 10, 9, 8, 13, 11, 14, 15, 27, 23, 21, 19, 20, 18, 17, 16, 28, 25, 24, 22, 29, 26, 30, 31, 58, 53, 48, 46, 44, 41, 40, 38, 43, 39, 37, 35, 36, 34, 33, 32, 59, 54, 52, 49, 51, 47, 45, 42, 60, 56, 55, 50, 61, 57, 62, 63, 121, 113, 108

Offset: 0

Antti Karttunen, Dec 18 2013

This permutation is obtained by "entangling" even and odd numbers with complementary pair A055938 & A005187, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A055938(n) to the parent node containing n, and each child to the right is obtained as A005187(n+1):
                                     0
                                     |
                  ...................1...................
                 2                                       3
       5......../ \........4                   6......../ \........7
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  12       10          9       8          13       11         14       15
27  23   21  19      20 18   17 16      28  25   24  22     29  26   30  31
etc.
For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to zero at the root from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.
Permutation A233276 gives the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233277.
Cf. A005187, A054429, A055938, A256991, A256478, A256479.
Cf. also A070939 (the binary width of both n and a(n)).
Related arrays: A255555, A255557.
Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).
As a composition of related permutations:
a(n) = A233276(A054429(n)).

Name changed and the illustration of binary tree added by Antti Karttunen, Apr 19 2015

A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

Original entry on oeis.org

1, 2, 3, 4, 7, 4, 10, 8, 12, 8, 18, 8, 22, 12, 15, 16, 31, 12, 34, 16, 25, 20, 41, 16, 39, 24, 34, 24, 53, 16, 56, 32, 42, 32, 49, 24, 70, 36, 48, 32, 78, 24, 81, 40, 48, 44, 88, 32, 84, 40, 63, 48, 101, 36, 79, 48, 72, 56, 112, 32, 116, 60, 69, 64, 98, 40, 130, 64, 90, 48, 137, 48, 142, 72, 81, 72, 121, 48, 152, 64

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Sequence differs from A035532 for the first time at n = 15, 21, 25, 27, 33, 35, 51, etc., i.e., at those composite n where A297115 has a nonzero value. - Antti Karttunen & M. F. Hasler, Mar 10 2018

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

Cf. A000010, A005187, A008683, A297108, A297110, A297114, A297115, A297117, A300244, A300723, A300724, A300725.

Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));

a(n) = Sum_{d|n} A005187(d)*A008683(n/d).
a(n) = n + A297114(n).
From Antti Karttunen, Mar 11 2018: (Start)
Sum A005187(n) x^n = Sum a(n)*x^n/(1-x^n). [Another way of saying that this is the Möbius transform of A005187. This was originally included in A035532 by mistake.]
a(n) = 2*phi(n) - A297115(n) = phi(n) + A297117(n).
a(n) = A005187(n) - A300244(n).
a(1) = 1; for n > 1, a(n) = A300723(n) + 2*A300724(n).
(End)

A101925 a(n) = A005187(n) + 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, 129

Offset: 0

Ralf Stephan, Dec 28 2004

nonn

Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.

p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
F. Ruskey, C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

Bisection of A089279. First differences are in A001511.
Cf. A000120, A005187, A046699.
Cf. A032184, A052278, A060055, A066318, A088229, A101926.

Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)
Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* Michael De Vlieger, Feb 04 2017 *)

PARI
```
a(n)=1+sum(k=1, n, valuation(k,2)+1)
```

a(n)=if(n==0,1,if((n%2)==0,2*a(n/2)+subst(Pol(binary(n)),x,1)-1,a(n-1)+1))

a(n)=2*n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022
(Python 3.10+)
def A101925(n): return (n<<1)-n.bit_count()+1 # Chai Wah Wu, Jul 13 2022

Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.

G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

cp's OEIS Frontend

A055938 Integers not generated by b(n) = b(floor(n/2)) + n (complement of A005187).

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A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

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A056982 a(n) = 4^A005187(n). The denominators of the Landau constants.

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A213714 Inverse function for injection A005187.

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A233275 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with even and odd numbers.

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A233276 a(0)=0, a(1)=1, after which a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).

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A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

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A233278 a(0)=0, a(1)=1, after which a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).