A000120 -id:A000120 - cp's OEIS frontend

A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

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

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 15 2001

nonn

Number of conjugacy classes in the symmetric group S_n that have odd number of elements.
Also sequence A001316 doubled.
Number of even numbers whose binary expansion is a child of the binary expansion of n. - Nadia Heninger and N. J. A. Sloane, Jun 06 2008
First differences of A151566. Sequence gives number of toothpicks added at the n-th generation of the leftist toothpick sequence A151566. - N. J. A. Sloane, Oct 20 2010
The Fi1 and Fi1 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal this sequence. - Johannes W. Meijer, Jun 05 2011
Also number of odd entries in n-th row of triangle of Stirling numbers of the first kind. - Istvan Mezo, Jul 21 2017

			a(3) = 2 because in S_3 there are two conjugacy classes with odd number of elements, the trivial conjugacy class and the conjugacy class of transpositions consisting of 3 elements: (12),(13),(23).
From _Omar E. Pol_, Oct 12 2011 (Start):
Written as a triangle:
1,
1,
2,2,
2,2,4,4,
2,2,4,4,4,4,8,8,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,4,4,8,8,8,8,16,16,8,...
(End)

I. G. MacDonald: Symmetric functions and Hall polynomials Oxford: Clarendon Press, 1979. Page 21.

Indranil Ghosh, Table of n, a(n) for n = 0..65536 (terms 0..1000 from Harry J. Smith)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A139251, A151566, A160407.

a000120:=func< n | &+Intseq(n, 2) >; [ 2^a000120(Floor(n/2)): n in [0..100] ]; // Klaus Brockhaus, Oct 15 2010

A060632 := proc(n) local k; add(binomial(n,2*k) mod 2, k=0..floor(n/2)); end: seq(A060632(n),n=0..94); # edited by Johannes W. Meijer, May 28 2011
A060632 := n -> 2^add(i, i = convert(iquo(n,2), base, 2)); # Peter Luschny, Jun 30 2011
A060632 := n -> igcd(2^n, n! / iquo(n,2)!^2);  # Peter Luschny, Jun 30 2011

a[n_] := 2^DigitCount[Floor[n/2], 2, 1]; Table[a[n], {n, 0, 94}] (* Jean-François Alcover, Feb 25 2014 *)

for (n=0, 1000, write("b060632.txt", n, " ", sum(k=0, floor(n/2), binomial(n, 2*k) % 2)) ) \\ Harry J. Smith, Sep 14 2009

a(n)=2^hammingweight(n\2) \\ Charles R Greathouse IV, Feb 06 2017

def A060632(n):
    return 2**bin(n/2)[2:].count("1") # Indranil Ghosh, Feb 06 2017

a(n) = sum{k=0..floor(n/2), C(n, 2k) mod 2} - Paul Barry, Jan 03 2005, Edited by Harry J. Smith, Sep 15 2009
a(n) = gcd(A056040(n), 2^n). - Peter Luschny, Jun 30 2011
G.f.: (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019

More terms from James Sellers, Apr 16 2001
Edited by N. J. A. Sloane, Jun 06 2008; Oct 11 2010
a(0) = 1 added by N. J. A. Sloane, Sep 14 2009
Formula corrected by Harry J. Smith, Sep 15 2009

A213724 Largest natural number x such that x = n + A000120(x), or zero if no such number exists.

Original entry on oeis.org

1, 3, 0, 5, 7, 0, 0, 9, 11, 0, 13, 15, 0, 0, 0, 17, 19, 0, 21, 23, 0, 0, 25, 27, 0, 29, 31, 0, 0, 0, 0, 33, 35, 0, 37, 39, 0, 0, 41, 43, 0, 45, 47, 0, 0, 0, 49, 51, 0, 53, 55, 0, 0, 57, 59, 0, 61, 63, 0, 0, 0, 0, 0, 65, 67, 0, 69, 71, 0, 0, 73, 75, 0, 77, 79, 0, 0, 0, 81, 83, 0, 85, 87, 0, 0, 89, 91

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

One more than A213723 at its nonzero points. a(A005187(n)) = A005408(n). Used for computing A213725-A213727.

Cf. A179016. - M. F. Hasler, Nov 01 2012

a213724 0 = 1
a213724 n = a213723 n + signum (a213723 n)
-- Reinhard Zumkeller, May 01 2015

(define (A213724 n) (if (zero? n) 1 (let ((v (A213723 n))) (if (zero? v) v (+ 1 v)))))

a(0)=1; and for n>=1, a(n)=0 if A079559(n)=0 (i.e., if A213723(n)=0); otherwise A213723(n)+1.

A147610 a(n) = 3^(wt(n-1)-1), where wt() = A000120().

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 9, 1, 3, 3, 9, 3, 9, 9, 27, 1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 3, 9, 9, 27, 9, 27, 27, 81, 9

Offset: 2

N. J. A. Sloane, Apr 29 2009

nonn

a(n) = A147582(n)/4.

			When written as a triangle:
.1,
.1,3,
.1,3,3,9,
.1,3,3,9,3,9,9,27,
.1,3,3,9,3,9,9,27,3,9,9,27,9,27,27,81,
.1,3,3,9,3,9,9,27,3,9,9,27,9,27,27,81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,
....
Rows converge to A048883. Row sums give A000302. Partial sums give A151920.

Omar E. Pol, Illustration of initial terms (Overlapping squares) [From _Omar E. Pol_, Nov 15 2009]

Cf. A048883, A000120, A000302, A151920, A147582, A048881.

Cf. A079314. - Omar E. Pol, Nov 15 2009

A000120 := proc(n) local a,d; a := 0 ; for d from 0 to ilog2(n) do a := a+ ( floor(n/2^d) mod 2) ; od: a ; end: A048881 := proc(n) A000120(n+1)-1 ; end: A147610 := proc(n) 3^A048881(n) ; end: seq(A147610(n),n=0..100) ; # R. J. Mathar, Apr 30 2009

a[n_] := 3^(DigitCount[n - 1, 2, 1] - 1);
a /@ Range[2, 100] (* Jean-François Alcover, Mar 24 2020 *)

a(n) = 3^(hammingweight(n-1)-1); \\ Michel Marcus, Mar 24 2020

a(n) = 3^A048881(n-2). - R. J. Mathar, Apr 30 2009
Recurrence: Write n = 2^i + 1 + j, 0 <= j < 2^i. Then a(2^i+1) = 1; for j>0, a(2^i+j+1) = 3*a(j+1). - N. J. A. Sloane, Jun 09 2009
G.f.: x*(Product_{k>=0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Extended by R. J. Mathar, Apr 30 2009
Offset corrected by N. J. A. Sloane, Jun 09 2009
Further edited by N. J. A. Sloane, Aug 06 2009

A151920 a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 21, 22, 25, 28, 37, 40, 49, 58, 85, 86, 89, 92, 101, 104, 113, 122, 149, 152, 161, 170, 197, 206, 233, 260, 341, 342, 345, 348, 357, 360, 369, 378, 405, 408, 417, 426, 453, 462, 489, 516, 597, 600, 609, 618, 645, 654, 681, 708, 789, 798, 825, 852, 933, 960

Offset: 0

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Partial sums of A147610 (but with offset changed to 0).

It appears that the first bisection gives the positive terms of A147562. - Omar E. Pol, Mar 07 2015

			n=3: (3^1+3^1+3^2+3^1)/3 = 18/3 = 6.
n=18: the binary expansion of 18+1 is 10011, i.e., 19 = 2^4 + 2^1 + 2^0.
The exponents of these powers of 2 (4, 1 and 0) reoccur as exponents in the powers of 4: a(19) = 3^0 * [(4^4 - 1) / 3 + 1] + 3^1 * [(4^1 - 1) / 3 + 1] + 3^2 * [(4^0 - 1)/3 + 1] = 1 * 86 + 3 * 2 + 9 * 1 = 101. - _David A. Corneth_, Mar 21 2015

Michael De Vlieger, Table of n, a(n) for n = 0..10000
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. 31.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000120, A130665, A147562, A147610, A153000, A160410, A160414.

Cf. A139250, A151917, A152998.

t = Nest[Join[#, # + 1] &, {0}, 14]; Table[Sum[3^t[[i + 1]], {i, 1, n}]/3, {n, 60}] (* Michael De Vlieger, Mar 21 2015 *)

a(n) = sum(i=1, n+1, 3^hammingweight(i))/3; \\ Michel Marcus, Mar 07 2015

a(n)={b=binary(n+1);t=#b;e=-1;sum(i=1,#b,e+=(b[i]==1);(b[i]==1)*3^e*((4^(#b-i)-1)/3+1))} \\ David A. Corneth, Mar 21 2015

a(n) = (A147562(n+2) - 1)/4 = (A151917(n+2) - 1)/2. - Omar E. Pol, Mar 13 2011
a(n) = (A130665(n+1) - 1)/3. - Omar E. Pol, Mar 07 2015
a(n) = a(n-1) + 3^A000120(n+1)/3. - David A. Corneth, Mar 21 2015

A228088 Numbers n for which there is a unique k which satisfies n = k + wt(k), where wt(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

0, 2, 3, 7, 8, 9, 10, 11, 12, 16, 20, 24, 25, 26, 27, 28, 29, 34, 35, 40, 41, 42, 43, 44, 45, 49, 53, 57, 58, 59, 60, 61, 62, 65, 66, 68, 69, 72, 73, 74, 75, 76, 77, 81, 85, 89, 90, 91, 92, 93, 94, 99, 100, 105, 106, 107, 108, 109, 110, 114, 118, 122, 123, 124

Offset: 1

Antti Karttunen, Aug 09 2013

wt(k) = A000120(k) is also called bitcount(k).
In other words, the positions of ones in A228085.
Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in exactly one way. - Matthew C. Russell, Oct 08 2013

			0 is in this sequence because there is a unique k such that k+A000120(k)=0, in this case k=0.
1 is not in this sequence because there is no such k that k+A000120(k) would be 1. (Instead 1 is in A010061).
2 is in this sequence because there is exactly one k that satisfies k+A000120(k)=2, namely k=1.
3 is in this sequence because there is exactly one k that satisfies k+A000120(k)=3, namely k=2.
4 is not in this sequence because there is no such k that k+A000120(k) would be 4. (Instead 4 is in A010061.)
5 is not in this sequence because there is more than one k that satisfies k+A000120(k)=5, namely k=3 and k=4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Subset of A228082.
Cf. A228089 (corresponding k's for each a(n)).
Cf. A228090 (the same k's sorted into ascending order).
Cf. A000120, A228085, A092391, A010061, A010062, A230091, A230092, A230058.
Cf. A227915.

a228088 n = a228088_list !! (n-1)
a228088_list = 0 : filter ((== 1) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013

a(n) = A092391(A228089(n)). [Consequence of the definitions of A228088 & A228089. Use the given Scheme-code to actually compute the sequence]

A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 2, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 7, 16, 17, 18, 19, 7, 20, 21, 22, 23, 24, 1, 25, 26, 27, 28, 29, 30, 17, 31, 32, 33, 34, 10, 35, 36, 37, 38, 39, 40, 41, 5, 42, 23, 43, 44, 45, 46, 47, 48, 49, 50, 51, 1, 52, 18, 53, 54, 55, 56, 57, 58, 59, 60, 46, 9, 61, 23, 62, 63, 39, 64, 65, 66, 67, 68, 69, 56, 70, 71, 72, 73, 47, 74

Offset: 1

Antti Karttunen, Aug 25 2018

Restricted growth sequence transform of ordered pair [A000120(n), A033879(n)], or equally, of ordered pair [A000120(n), A294898(n)].
For all i, j:
  A318311(i) = A318311(j) => a(i) = a(j),
  a(i) = a(j) => A286449(i) = A286449(j),
  a(i) = a(j) => A294898(i) = A294898(j).
In the scatter plot one can see the effects of both base-2 related A000120 (binary weight of n) and prime factorization related A033879 (deficiency of n) graphically mixed: from the former, a square grid pattern, and from the latter the black rays that emanate from the origin. The same is true for A323898, while in the ordinal transform of this sequence, A331184, such effects are harder to visually discern. - Antti Karttunen, Jan 13 2020

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

Cf. A000120, A033879, A286449, A295881, A295882, A294898.
Cf. A318311, A323889, A323892, A323898, A324344, A324380, A324390 for similar constructions.
Cf. A331184 (ordinal transform).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];
v318310 = rgs_transform(vector(up_to,n,A318310aux(n)));
A318310(n) = v318310[n];

Name changed by Antti Karttunen, Jan 13 2020

A094387 Numbers k such that gcd(k, A000120(k)) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 15, 16, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 52, 53, 56, 57, 59, 61, 62, 64, 65, 67, 70, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 97, 98, 99, 100, 101, 103

Offset: 1

Benoit Cloitre, Jun 08 2004

This sequence has density 6/Pi^2 (Olivier, 1975).

All primes, powers of 2, and powers of two plus one are terms of this sequence. - William Boyles, Jan 27 2022

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge University Press, 2003, p. 117.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Christian Mauduit, Carl Pomerance and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Cf. A000120, A059956.

Select[Range[100], CoprimeQ[#, DigitCount[#, 2, 1]] &] (* Amiram Eldar, Nov 22 2020 *)

lista(nn) = {for (n=1, nn, if (gcd(n, norml2(binary(n))) == 1, print1(n, ", ")););} \\ Michel Marcus, May 25 2013

from math import gcd
def ok(n): return gcd(n, bin(n).count('1')) == 1
print([k for k in range(104) if ok(k)]) # Michael S. Branicky, Jan 25 2022

A175524 A000120-deficient numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Vladimir Shevelev, Dec 03 2010

For a more precise definition, see comment in A175522.
All odd primes (A065091) are in the sequence. Squares of the form (2^n+3)^2, n>=3, where 2^n+3 is prime (A057733), are also in the sequence. [Proof: (2^n+3)^2 = 2^(2*n)+2^(n+2)+2^(n+1)+2^3+1. Thus, since n>=3, A000120((2^n+3)^2)=5. Also, for primes of the form 2^n+3, (2^n+3)^2 has only two proper divisors, 1 and 2^n+3, so A000120(1)+A000120(2^n+3) = 4, and in conclusion, (2^n+3)^2 is deficient. QED]
It is natural to assume that there are infinitely many primes of the form 2^n+3 (by analogy with the Mersenne sequence 2^n-1). If this is true, the sequence contains infinitely many composite numbers, because it contains all of the form (2^n+3)^2.
a(n) = A006005(n) for n <= 30;

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A175522 (perfect version), A175526 (abundant version), A000120, A005100, A005101, A006005, A192895.

import Data.List (findIndices)
a175524 n = a175524_list !! (n-1)
a175524_list = map (+ 1) $ findIndices (< 0) a192895_list
-- Reinhard Zumkeller, Jul 12 2011

Select[Range[270], DivisorSum[#, DigitCount[#, 2, 1] &] < 2*DigitCount[#, 2, 1] &] (* Amiram Eldar, Jul 25 2023 *)

is(n)=sumdiv(n,d,hammingweight(d))<2*hammingweight(n) \\ Charles R Greathouse IV, Jan 28 2016

is_A175524 = lambda x: sum(A000120(d) for d in divisors(x)) < 2*A000120(x)
A175524 = filter(is_A175524, IntegerRange(1, 10**4)) # D. S. McNeil, Dec 04 2010

A192895(a(n)) < 0. - Reinhard Zumkeller, Jul 12 2011

More terms from Amiram Eldar, Feb 18 2019

A295296 Numbers n such that the sum of their divisors + the number of ones in their binary expansion = 2n; numbers for which A000203(n) + A000120(n) = 2n.

Original entry on oeis.org

1, 2, 3, 4, 8, 10, 16, 32, 64, 128, 136, 256, 315, 512, 1024, 2048, 4096, 8192, 16384, 32768, 32896, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 2147516416

Offset: 1

Antti Karttunen, Nov 26 2017

Numbers n such that their binary weight is equal to their deficiency.
Numbers n such that A000120(n) = A033879(n), or equally A000203(n) = A005187(n), or equally A001065(n) = A011371(n).
2^(2^k-1) * (2^(2^k) + 1) is in the sequence if and only if (2^(2^k) + 1) is a (Fermat) prime (A019434) which as of today is known to be the case for 0 <= k <= 4 giving the terms 3, 10, 136, 32896 and 2147516416. - David A. Corneth, Nov 26 2017
It would be nice to know whether 315 is the only term that is neither in A191363 nor a power of two.
Any term that is either a square or twice a square (in A028982) must be odious (in A000069), and vice versa.
If there's an odd term below 10^30 besides 315 then it must be divisible by a prime >= 23. - David A. Corneth, Nov 27 2017
221753180448460815 is odd and also a term of this sequence. - Alexander Violette, Dec 24 2020

			A000203(315) = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 315 = 624. 2*315 - 624 = 6, and when 315 is written in binary, 100111011, we see that it has six 1-bits. Thus 315 is included in the sequence.

Max Alekseyev, Table of n, a(n) for n = 1..70 (first 52 terms from Giovanni Resta)
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Positions of zeros in A294898 and A294899.
Subsequence of A005100 and also of A295298.
Subsequences: A000079, A191363 (the five known terms).
Cf. A000120, A000203, A001065, A005187, A011371, A019434, A033879, A141548.

Select[Range[2^20], DivisorSigma[1, #] + DigitCount[#, 2, 1] == 2 # &] (* Michael De Vlieger, Nov 26 2017 *)

for(n=1, oo, if(sigma(n)+hammingweight(n) == 2*n, print1(n, ", ")));

Terms a(35) and beyond from Giovanni Resta, Feb 27 2020

A371176 Numbers k such that A000120(k) <= A001511(k).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 32, 34, 36, 40, 44, 48, 52, 56, 64, 66, 68, 72, 76, 80, 84, 88, 96, 100, 104, 112, 120, 128, 130, 132, 136, 140, 144, 148, 152, 160, 164, 168, 176, 184, 192, 196, 200, 208, 216, 224, 232, 240, 256, 258, 260, 264, 268

Offset: 1

Mikhail Kurkov, Mar 14 2024

It appears that this sequence is obtained when ordering Schreier sets as explained in the Bird link. See decM(n) PARI code. - Michel Marcus, May 31 2024
That is correct since the binary representation of these numbers can be put into 1-to-1 correspondence with Schreier sets, which satisfy |X| <= min X, using the indicator function of X as the bits (starting from the right, LSB). The reason is that A000120 then computes |X| and A001511 computes min X. For example, the Schreier set X = {2, 5} can be mapped to 10010_2 = 18. - Michael S. Branicky, May 31 2024
From David A. Corneth, May 31 2024: (Start)
If k is in the sequence then so is 2*k.
a(A000045(k)) = 2^(k-2) for k >= 2. (End)
Apart from a(1), all terms are even. - Paolo Xausa, May 31 2024
Zeckendorf representation of n with rewrite 0 -> 0, {0, 1} -> 1 and k-1 zeros appended to the right side (where k is the number of ones in the given representation) and then interpreted as binary expansion is the same as a(n) (see the first formula). - Mikhail Kurkov, Oct 21 2024

Michel Marcus, Table of n, a(n) for n = 1..10945
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Vaclav Kotesovec, Plot of log(a(n))/log(n) for n = 2..2129243
Vaclav Kotesovec, Plot of a(n) / 2^(log(sqrt(5)*n)/log((1 + sqrt(5))/2) - 2) for n = 2..2129243

Cf. A000045, A000120, A001316, A001511, A003849, A005206, A007895, A048679, A066628, A072649, A213911.
Cf. A355489, A373345, A373347 (complement), A373557.
Cf. A104287.

filter:= proc(n) convert(convert(n,base,2),`+`) <= 1+padic:-ordp(n,2) end proc:
select(filter, [1,seq(i,i=2..1000,2)]); # Robert Israel, Oct 20 2024

Join[{1}, Select[Range[2, 1000, 2], DigitSum[#, 2] <= IntegerExponent[#, 2] + 1 &]] (* Paolo Xausa, Aug 12 2025 *)

isok(n) = hammingweight(n) <= (valuation(n, 2) + 1)

M(n) = my(list=List()); for (i=1, n, forsubset(i, s, my(bOk = if (#s && (vecmax(s) == n), #s <= vecmin(s), 0)); if (bOk, listput(list, vecsort(Vec(s),,4))););); Vec(list);
decM(nn) = my(v = vector(nn, k, M(k)), list=List()); for (i=1, #v, my(vi = v[i]); for (j=1, #vi, my(s = vecsort(vi[j]), slist=List(), m = vecmax(s)); forstep(k=m, 1, -1, listput(slist, sign(vecsearch(s, k)))); listput(list, fromdigits(Vec(slist), 2)););); vecsort(Vec(list)); \\ Michel Marcus, May 31 2024

def ok(n): return n.bit_count() <= (-n&n).bit_length()
print([k for k in range(1, 300) if ok(k)]) # Michael S. Branicky, May 31 2024

# Assuming the list starts with 0.
def a():
    n = na = nb = 1
    while True:
        yield not(nb < (na - 1) << 1)
        nb, na = na, n.bit_count()
        n += 1
aList = a(); print([n for n in range(77) if next(aList)]) # Peter Luschny, Jun 07 2024

a(n) = b(n)*A001316(b(n))/2 where b(n) = A048679(n).
a(n) = Sum_{i=0..n-1} 2^A213911(i).
a(n) = 2^(A072649(n) - 1) + [c(n) > 0]*2*a(c(n)) where c(n) = A066628(n).
a(n) = 2*a(A005206(n)) - A003849(n)*2^A007895(n-1) for n > 1 with a(1) = 1.
Conjecture: lim sup_{n -> oo} log(a(n))/log(n) = log(2) / log((1 + sqrt(5))/2) = 1.440420090412556479... = A104287. - Vaclav Kotesovec, Aug 12 2025

cp's OEIS Frontend

A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

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A213724 Largest natural number x such that x = n + A000120(x), or zero if no such number exists.

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A147610 a(n) = 3^(wt(n-1)-1), where wt() = A000120().

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A151920 a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().

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A228088 Numbers n for which there is a unique k which satisfies n = k + wt(k), where wt(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

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A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

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A094387 Numbers k such that gcd(k, A000120(k)) = 1.

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