A053645 -id:A053645 - cp's OEIS frontend

A096111 If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).

1, 2, 2, 3, 3, 6, 6, 4, 4, 8, 8, 12, 12, 24, 24, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 40, 40, 60, 60, 120, 120, 6, 6, 12, 12, 18, 18, 36, 36, 24, 24, 48, 48, 72, 72, 144, 144, 30, 30, 60, 60, 90, 90, 180, 180, 120, 120, 240, 240, 360, 360, 720, 720, 7, 7, 14, 14, 21, 21

Offset: 0

Amarnath Murthy, Jun 29 2004

nonn

Each n > 1 occurs 2*A045778(n) times in the sequence.
f(n+2^k) = (k+1)*f(n) if 2^k > n+1. - Robert Israel, Apr 25 2016
If the binary indices of n (row n of A048793) are the positions 1's in its reversed binary expansion, then a(n) is the product of all binary indices of n + 1. The number of binary indices of n is A000120(n), their sum is A029931(n), and their average is A326699(n)/A326700(n). - Gus Wiseman, Jul 27 2019

Peter Kagey, Table of n, a(n) for n = 0..10000

Permutation of A096115, i.e. a(n) = A096115(A122198(n+1)) [Note the different starting offsets]. Bisection: A121663. Cf. A096113, A052330.
Cf. A029931.
Cf. A000120, A034797, A048793, A070939, A291166, A326031, A326672, A326673.

f:= proc(n) local L;
    L:= convert(2*n+2,base,2);
    convert(subs(0=NULL,zip(`*`,L, [$0..nops(L)-1])),`*`);
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016

CoefficientList[(Product[1 + k x^(2^(k - 1)), {k, 7}] - 1)/x, x] (* Michael De Vlieger, Apr 08 2016 *)
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];Table[Times@@bpe[n+1],{n,0,100}] (* Gus Wiseman, Jul 26 2019 *)

N=166; q='q+O('q^N);
gf= (prod(n=1,1+ceil(log(N)/log(2)), 1+n*q^(2^(n-1)) ) - 1) / q;
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

(define (A096111 n) (cond ((pow2? (+ n 1)) (+ 2 (A000523 n))) (else (* (+ 1 (A000523 n)) (A096111 (A053645 n))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))

G.f.: ( prod(k>=1, 1+k*x^(2^(k-1)) )- 1 ) / x. - Vladeta Jovovic, Nov 08 2005

a(n) is the product of the exponents in the binary expansion of 2*n + 2. - Peter Kagey, Apr 24 2016

Edited, extended and Scheme code added by Antti Karttunen, Aug 25 2006

A126441 Tabular arrangement of the natural numbers: the row on which any nonzero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's.

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 7, 8, 9, 6, 11, 0, 0, 0, 15, 16, 17, 10, 19, 0, 13, 0, 23, 0, 0, 0, 0, 0, 0, 0, 31, 32, 33, 18, 35, 12, 21, 14, 39, 0, 0, 0, 27, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 64, 65, 34, 67, 20, 37, 22, 71, 0, 25, 0, 43, 0, 29, 0, 79, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0

Offset: 0

Alford Arnold, Jan 19 2007

Note: 1 might be a more natural starting offset for this sequence, although the identities concerning A053645 and A161511 would have to be changed. - Antti Karttunen, Oct 12 2009.
This can be regarded as an arrangement of the partitions, indexed by position in A125106. The partitions in a given row all have the same remaining partition when the largest part is removed; specifically, the partition indexed by the row number in A125106 (with row 0 having the empty partition remaining).
The first value on row n is A004760(n+1). The second value on each row is A004760(n+1) plus A062383(n); subsequent values increase by ever enlarging powers of two. Or equivalently, each subsequent value on the row after the first nonzero value is given by A004754(previous value on the same row).
A055941(r) tells how many terms the row r (>= 0) has been shifted rightward from its "natural position", i.e. with how many zeros that row has been prepended.
The number of (nonzero) entries in column k is A000041(k).

			The largest power of 2 <= 6 is 4, 6 - 4 = 2, so 6 is in row 2. By A125106, 6 corresponds to the partition [2^2], total 4, so 6 goes in column 4. Thus T(2,4) = 6.
The table begins:
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275

A. Karttunen, Table of n, a(n) for n = 0..65534 (first 16 columns)

Cf. A125106, A053645, A000041, A004760, A062383, A000079 (column lengths).
A053645(a(A166274(n))) = A053645(1+A166274(n)) for all n>=1.
Positions of zeros: A166275, this sequence without zeros: A161924. A161920(n) gives the position of the first nonzero term on the row n-1.

columns = 7; row[n_] := n-2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n-1)/2]+1]; Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[T[n-1, k], {k, 1, columns}, {n, 1, 2^(k-1)}] // Flatten (* Jean-François Alcover, Sep 09 2017 *)

Edited by Franklin T. Adams-Watters, Jan 23 2007

Further edited and Scheme-code added by Antti Karttunen, Oct 12 2009

A121663 a(0) = 1; if n = 2^k, a(n) = k+2, otherwise a(n)=(A000523(n)+2)*a(A053645(n)).

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 6, 12, 18, 36, 24, 48, 72, 144, 30, 60, 90, 180, 120, 240, 360, 720, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 42, 84, 126, 252, 168, 336, 504, 1008, 210, 420, 630, 1260, 840, 1680

Offset: 0

Antti Karttunen, Aug 25 2006

nonn

Each n occurs A045778(n) times in the sequence.

Ivan Neretin, Table of n, a(n) for n = 0..8192

Bisection of A096111.

f[0] := 1; f[n_] := If[(b = n - 2^(k = Floor[Log2[n]])) == 0, k + 2, (k + 2)*f[b]]; Table[f[n], {n, 0, 61}] (* Ivan Neretin, May 09 2015 *)

(define (A121663 n) (cond ((zero? n) 1) ((pow2? n) (+ 2 (A000523 n))) (else (* (+ 2 (A000523 n)) (A121663 (A053645 n))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))

G.f.: Product_{k>=0} (1 + (k + 2) * x^(2^k)). - Ilya Gutkovskiy, Aug 19 2019

A300725 Möbius transform of A053645(n), distance to the largest power of 2 less than or equal to n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 0, 0, 1, 3, 2, 5, 3, 5, 0, 1, 0, 3, 2, 1, 3, 7, 4, 8, 5, 10, 6, 13, 5, 15, 0, -3, 1, -1, 0, 5, 3, 1, 4, 9, 1, 11, 6, 6, 7, 15, 8, 14, 8, 17, 10, 21, 10, 19, 12, 21, 13, 27, 10, 29, 15, 26, 0, -5, -3, 3, 2, -3, -1, 7, 0, 9, 5, -4, 6, 7, 1, 15, 8, 6, 9, 19, 2, 19, 11, 9, 12, 25, 6, 19, 14, 13, 15, 27

Offset: 1

Antti Karttunen, Mar 11 2018

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

Cf. A000010, A008683, A053644, A053645, A297111, A297115, A300723, A300724.

With[{s = Array[# - 2^Floor@ Log2@ # &, 95]}, Table[DivisorSum[n, MoebiusMu[n/#] s[[#]] &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A300725(n) = sumdiv(n,d,moebius(n/d)*A053645(d));

a(n) = Sum_{d|n} A008683(n/d)*A053645(d).

a(n) + A300724(n) = A000010(n).

A300723 Möbius-transform of A005187(A053645(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 4, 0, 0, 2, 4, 4, 8, 6, 9, 0, 1, 0, 4, 4, 3, 6, 11, 8, 15, 10, 18, 12, 23, 10, 26, 0, -4, 2, -1, 0, 8, 6, 2, 8, 16, 2, 19, 12, 12, 14, 26, 16, 28, 16, 33, 20, 39, 20, 37, 24, 42, 26, 50, 20, 54, 30, 49, 0, -8, -6, 4, 4, -4, -2, 11, 0, 16, 10, -7, 12, 15, 2, 26, 16, 13, 18, 35, 4, 37, 22, 18, 24, 47, 12

Offset: 1

Antti Karttunen, Mar 12 2018

sign

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

Cf. A005187, A008683, A053645, A297111, A300724, A300725.

With[{s = Array[2 # - DigitCount[2 #, 2, 1] &[# - 2^Floor@ Log2@ #] &, 90]}, Table[DivisorSum[n, MoebiusMu[n/#] s[[#]] &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A300723(n) = sumdiv(n,d,moebius(n/d)*A005187(A053645(d)));

a(n) = Sum_{d|n} A008683(n/d)*A005187(A053645(d)).

a(1) = 0; for n > 1, a(n) = A297111(n) - 2*A300724(n).

A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 08 2019

Also the restricted growth sequence transform of function f(1) = 0, f(n) = [A053645(n), A278222(n)] for n > 1.
For all i, j:
  a(i) = a(j) => A286622(i) = A286622(j),
  a(i) = a(j) => A323235(i) = A323235(j),
  a(i) = a(j) => A323236(i) = A323236(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A053645, A079944, A278222, A286622, A323235, A323236.

Cf. also A300226 (an analogous filter sequence for prime factorization).

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; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323234aux(n) = if(1==n,0,[A053645(n), A079944off0(n-2)]);
v323234 = rgs_transform(vector(up_to,n,A323234aux(n)));
A323234(n) = v323234[n];

A083741 a(n) = L(n) + a(L(n)), where L(n) = n - 2^floor(log_2(n)) (A053645).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 4, 0, 1, 2, 4, 4, 6, 8, 11, 0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 16, 18, 20, 23, 24, 27, 30, 34, 32, 35, 38, 42, 44, 48, 52, 57, 0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 16, 18, 20

Offset: 0

Ralf Stephan, May 05 2003

a(2^j)=0. Local extrema are a(2^j-1) = 2^j-j-1 (A000295).

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98 (1992), pp. 163-197, ex. 24. (PS file on author's web page.)

Cf. A000295, A272011, A298043.

f[l_]:=Join[l,l-1+Range[Length[l]]]; Nest[f,{0},7] (* Ray Chandler, Jun 01 2010 *)

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

a(n) = my(v=binary(n),c=-1); for(i=1,#v, if(v[i],v[i]=c++)); fromdigits(v,2); \\ Kevin Ryde, Apr 16 2024

a(0)=0, a(1)=0, a(2n)=2a(n), a(4n+1)=2a(n)+a(2n+1), a(4n+3)=-2a(n)+3a(2n+1)+1.

a(n) = Sum_{i=0..k} i*2^e[i] where the binary expansion of n is n = Sum_{i=0..k} 2^e[i] with decreasing exponents e[0] > ... > e[k] (A272011). - Kevin Ryde, Apr 16 2024

Extended by Ray Chandler, Mar 04 2010

A346422 a(n) = (1 + A014081(n))*a(A053645(n)) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 4, 2, 2, 6, 24, 1, 1, 1, 4, 1, 1, 4, 18, 2, 2, 2, 12, 6, 6, 24, 120, 1, 1, 1, 4, 1, 1, 4, 18, 1, 1, 1, 8, 4, 4, 18, 96, 2, 2, 2, 12, 2, 2, 12, 72, 6, 6, 6, 48, 24, 24, 120, 720, 1, 1, 1, 4, 1, 1, 4, 18, 1, 1, 1, 8, 4, 4, 18, 96

Offset: 0

Mikhail Kurkov, Aug 08 2021 [verification needed]

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16.

Cf. A003714 (positions of 1's), A014081, A036987, A053645.

Nest[Append[#1, (1 + Count[Partition[IntegerDigits[#2, 2], 2, 1], {1, 1}]) #1[[#2 - 2^Floor@ Log2[#2] + 1]]] & @@ {#, Length[#]} &, {1}, 79] (* Michael De Vlieger, Feb 04 2022 *)

a(n)=if(n==0, 1, (1+b(n))*a(c(n)))
b(n)=if(n==1, 0, if(n%4<2, b(n\4), b(n\2) + n%2)) \\ A014081
c(n)=if(n==1, 0, 2*c(n\2) + n%2) \\ A053645

a(n) = my(f=1,ret=1); if(n, for(i=0,logint(n,2), if(bittest(n,i), ret*=(f+=bittest(n,i-1))))); ret; \\ Kevin Ryde, Aug 25 2021

from functools import lru_cache
from re import split
@lru_cache(maxsize=None)
def A346422(n): return 1 if n <= 1 else A346422(int((s:= bin(n)[2:])[1:],2))*(1+sum(len(d)-1 for d in split('0+', s) if d != '')) # Chai Wah Wu, Feb 04 2022

a(n) = (1 + A014081(n))*a(A053645(n)) for n > 0 with a(0) = 1.
a(4n+1) = a(2n) = a(n) for n > 0 with a(0) = a(1) = 1.
a(4n+3) = 2!*b(n), a(8n+11) = 2*2!*b'(n), b'(2n) = b(n), b'(2n+1) = b'(n),
a(8n+7) = 3!*c(n), a(16n+23) = 3*3!*c'(n), c'(2n) = c(n), c'(2n+1) = c'(n),
a(16n+15) = 4!*d(n), a(32n+47) = 4*4!*d'(n), d'(2n) = d(n), d'(2n+1) = d'(n),
a(32n+31) = 5!*e(n), a(64n+95) = 5*5!*e'(n), e'(2n) = e(n), e'(2n+1) = e'(n),
and so on, i.e.,
a(2^m*(n+1) - 1) = m!*z(n), a(2^m*(2n+3) - 1) = m*m!*z'(n), z'(2n) = z(n), z'(2n+1) = z'(n).
From that, we get:
a(2^3*(2n+1) + 3) = 2*a(4n+3), a(2^3*(2n+1) + 11) = a(8n+11),
a(2^4*(2n+1) + 7) = 3*a(8n+7), a(2^4*(2n+1) + 23) = a(16n+23),
a(2^5*(2n+1) + 15) = 4*a(16n+15), a(2^5*(2n+1) + 47) = a(32n+47),
a(2^6*(2n+1) + 31) = 5*a(32n+31), a(2^6*(2n+1) + 95) = a(64n+95),
and so on, i.e.,
a(2^m*(4n+3) - 1) = m*a(2^m*(n+1) - 1), a(2^m*(4n+5) - 1) = a(2^m*(2n+3) - 1).
Let
p(n) = 0 if A036987(n) = 1 otherwise p(2n+1) = 2 + p(n), p(2n) = 2 - (n mod 2) for n > 0 with p(0) = p(1) = 0,
q(2n+1) = 2^(n+2) - 1, q(2n) = 2^(n+2) + q(2n-1) for n > 0 with q(1) = 3,
p_1(n) = 0 if A036987(n) = 1 otherwise q(p(n)) for n > 0 with p_1(0) = 0,
p_2(n) = 0 if A036987(n) = 1 otherwise p_2(2n+1) = p_2(n), p_2(2n) = floor((n - 1)/2) for n > 0 with p_2(0) = p_2(1) = 0,
so
a(4n+3) = (log_2(4n+4))! if A036987(4n+3) = 1 otherwise (1 + (p(n) mod 2)*(p(n) + 1)/2)*a(p_2(n)*2^(floor(p(n)/2) + 2) + p_1(n)) for n >= 0.
a((4^n - 1)/3) = 1 for n >= 0.
a(2^m*(2^n - 1)) = n! for n > 0, m >= 0.

A357990 Square array T(n, k), n >= 0, k > 0, read by antidiagonals, where T(0, k) = 1 for k > 0 and where T(n, k) = R(n, k+1) - R(n, k) for n > 0, k > 0. Here R(n, k) = T(A053645(n), k)*k^(A290255(n) + 1).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 7, 1, 7, 1, 1, 3, 19, 1, 9, 1, 1, 7, 5, 37, 1, 11, 1, 1, 1, 11, 7, 61, 1, 13, 1, 1, 15, 1, 15, 9, 91, 1, 15, 1, 1, 7, 65, 1, 19, 11, 127, 1, 17, 1, 1, 17, 19, 175, 1, 23, 13, 169, 1, 19, 1, 1, 3, 43, 37, 369, 1, 27, 15, 217, 1, 21

Offset: 0

Mikhail Kurkov, Nov 20 2022

			Square array begins:
   1,  1,   1,   1,   1,    1,    1,    1, ...
   1,  1,   1,   1,   1,    1,    1,    1, ...
   3,  5,   7,   9,  11,   13,   15,   17, ...
   1,  1,   1,   1,   1,    1,    1,    1, ...
   7, 19,  37,  61,  91,  127,  169,  217, ...
   3,  5,   7,   9,  11,   13,   15,   17, ...
   7, 11,  15,  19,  23,   27,   31,   35, ...
   1,  1,   1,   1,   1,    1,    1,    1, ...
  15, 65, 175, 369, 671, 1105, 1695, 2465, ...

Cf. A000120, A053645, A290255, A329369.

R(n,k)=my(L=logint(n, 2), A=n - 2^L); T(A, k)*k^(L - if(A>0, logint(A, 2) + 1) + 1)
T(n,k)=if(n==0, 1, R(n, k+1) - R(n, k))

T(n, k) = my(A = 2*n+1, B, C, v1, v2); v1 = []; while(A > 0, B=valuation(A, 2); v1=concat(v1, B+1); A \= 2^(B+1)); v1 = Vecrev(v1); A = #v1; v2 = vector(A, i, 1); for(i=1, A-1, B = A-i; for(j=1, B, C = B-j+k+1; v2[j] = v2[j]*C^v1[B] - v2[j+1]*(C-1)^v1[B])); v2[1] \\ Mikhail Kurkov, Apr 30 2024

Conjecture: T(n, 1) = A329369(n).

A075149 Sum_{i=0..2A053645(n)} (C(2A053645(n),i) mod 2)*A000045(n-i) [where C(r,c) is the binomial coefficient (A007318) and A000045(n) is the n-th Fibonacci number].

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 9, 21, 21, 47, 63, 141, 147, 329, 441, 987, 987, 2207, 2961, 6621, 6909, 15449, 20727, 46347, 46389, 103729, 139167, 311187, 324723, 726103, 974169, 2178309, 2178309, 4870847, 6534927, 14612541, 15248163, 34095929, 45744489

Offset: 0

Antti Karttunen, Sep 05 2002

nonn

A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

Bisection gives A050614.

with(combinat); [seq(A075149(n,n=0..50)]; A075149 := n -> add((binomial(2*r(n),i) mod 2)*fibonacci(n-i),i=0..2*r(n));
r := n -> n - 2^floor_log_2(n);
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;

cp's OEIS Frontend

A096111 If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).

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A126441 Tabular arrangement of the natural numbers: the row on which any nonzero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's.

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A121663 a(0) = 1; if n = 2^k, a(n) = k+2, otherwise a(n)=(A000523(n)+2)*a(A053645(n)).

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A300725 Möbius transform of A053645(n), distance to the largest power of 2 less than or equal to n.

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A300723 Möbius-transform of A005187(A053645(n)).

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A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

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A083741 a(n) = L(n) + a(L(n)), where L(n) = n - 2^floor(log_2(n)) (A053645).

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A346422 a(n) = (1 + A014081(n))*a(A053645(n)) for n > 0 with a(0) = 1.

A075149 Sum_{i=0..2A053645(n)} (C(2A053645(n),i) mod 2)*A000045(n-i) [where C(r,c) is the binomial coefficient (A007318) and A000045(n) is the n-th Fibonacci number].