A001511 -id:A001511 - cp's OEIS frontend

A092119 EULER transform of A001511.

1, 1, 3, 4, 10, 13, 26, 35, 66, 88, 150, 202, 331, 442, 688, 919, 1394, 1848, 2716, 3590, 5174, 6796, 9589, 12542, 17440, 22680, 31055, 40208, 54420, 70096, 93772, 120256, 159380, 203436, 267142, 339573, 442478, 560050, 724302, 913198, 1173375, 1473622

Offset: 0

Vladeta Jovovic, Mar 29 2004

nonn

From Gary W. Adamson, Feb 11 2010: (Start)
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),
A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),
A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. (End)
Let M = triangle A173238 as an infinite lower triangular matrix. Then A092119 = lim_{n->infinity} M^n. Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + ...), and A(x) = polcoeff A092119. Then P(x) = A(x) / A(x^2), an example of a conjectured infinite set of operations (cf. A173238). - Gary W. Adamson, Feb 13 2010

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
N. J. A. Sloane, Transforms

Cf. A000041. - Gary W. Adamson, Feb 11 2010

Cf. A173241.

# Uses EulerTransform from A358369.
t := EulerTransform(n -> padic[ordp](2*n, 2)):
seq(t(n), n = 0..41); # Peter Luschny, Nov 18 2022

m = 42;
1/Product[QPochhammer[x^(2^k)], {k, 0, Log[2, m]//Ceiling}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 14 2020, after Joerg Arndt *)

N=66; x='x+O('x^N); /* that many terms */
gf=1/prod(e=0,ceil(log(N)/log(2)),eta(x^(2^e)));
Vec(gf) /* show terms */ /* Joerg Arndt, Jun 21 2011 */

G.f.: 1/Product_{k>=0} P(x^(2^k)) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011

A287896 a(n) = A002487(n)*A001511(n).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 6, 5, 6, 5, 6, 4, 5, 5, 8, 7, 9, 8, 10, 7, 8, 7, 10, 8, 9, 7, 8, 5, 6, 6, 10, 9, 12, 11, 14, 10, 12, 11, 16, 13, 15, 12, 14, 9, 10, 9, 14, 12, 15, 13, 16, 11, 12, 10, 14, 11, 12, 9, 10, 6, 7, 7, 12, 11, 15, 14, 18, 13, 16, 15, 22, 18, 21, 17, 20, 13, 15, 14, 22, 19, 24, 21, 26, 18, 20, 17

Offset: 1

I. V. Serov, Jun 02 2017

nonn

Proposed name: N-fusc.
Each number n>0 appears in this sequence exactly n times.
From Yosu Yurramendi, Apr 08 2019: (Start)
The terms (n>0) may be written as a left-justified array with rows of length 2^m:
  1,
  2, 2,
  3, 3,  4,  3,
  4, 4,  6,  5,  6,  5,  6,  4,
  5, 5,  8,  7,  9,  8, 10,  7,  8,  7, 10,  8,  9,  7,  8,  5,
  6, 6, 10,  9, 12, 11, 14, 10, 12, 11, 16, 13, 15, 12, 14,  9, 10,  9, ...
  ...
as well as right-justified fashion:
                                                                      1,
                                                                   2, 2,
                                                            3, 3,  4, 3,
                                            4,  4,  6,  5,  6, 5,  6, 4,
             5, 5,  8,  7,  9,  8, 10,  7,  8,  7, 10,  8,  9, 7,  8, 5,
... 14,  9, 10, 9, 14, 12, 15, 13, 16, 11, 12, 10, 14, 11, 12, 9, 10, 6,
From these two dispositions interesting properties can be induced (see FORMULA section)
(End)

I. V. Serov, Table of n, a(n) for n = 1..8192

Cf. A001511, A002487, A007306, A288002.

Table[Block[{a = 1, b = 0, m = n}, While[m > 0, If[OddQ@ m, b = a + b, a = a + b]; m = Floor[m/2]]; b] IntegerExponent[2 n, 2], {n, 89}] (* Michael De Vlieger, Jun 14 2017, after Jean-François Alcover at A002487 *)

from functools import reduce
def A287896(n): return (n&-n).bit_length()*sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) # Chai Wah Wu, Jul 14 2022

a(1) = 1; for n>1: a(n) = (A002487(n-1) + A002487(n) + A002487(n+1))/2.
a(n) = A007306(n) - A288002(n).
From Yosu Yurramendi, Apr 08 2019: (Start)
For m >= 0, 0 <= k < 2^m, a(2^(m+1)+k) - a(2^m+k) = a(k). a(0) = 1 is needed.
For m >= 0, 0 <= k < 2^m, a(2^(m+1)-1-k) - a(2^(m)-1-k) = a(k).
(End)

A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 11, 24, 13, 20, 15, 10, 17, 26, 19, 34, 21, 32, 23, 48, 25, 38, 27, 54, 29, 44, 31, 12, 33, 50, 35, 64, 37, 56, 39, 76, 41, 62, 43, 84, 45, 68, 47, 120, 49, 74, 51, 94, 53, 80, 55, 90, 57, 86, 59, 114, 61, 92, 63, 16, 65, 98, 67, 124, 69, 104, 71, 118, 73, 110, 75, 144, 77, 116, 79, 142, 81

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

In the essence, a(n) tells which number in square array A249741 (the sieve of Eratosthenes minus 1) is at the same position where n is in array A135764, which is formed from odd numbers whose binary expansions are shifted successively leftwards on the successive rows. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.

Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A054582, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249812.
Cf. A000079, A000265, A001511, A003602, A005408, A006093, A007814, A054582, A083140, A083221, A249741.
Similar or related permutations: A249814 ("deep variant"), A246676, A249815, A114881, A209268, A249725, A249741.
Differs from A246676 for the first time at n=14, where a(14)=20, while
A246676(14)=26.

(define (A249811 n) (+ -1 (A083221bi (A001511 n) (A003602 n))))
;; Code for A083221bi given in A083221

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(n) = A083221(A001511(n),A003602(n)) - 1 = A249741(A001511(n),A003602(n)).
As a composition of related permutations:
a(n) = A114881(A209268(n)).
a(n) = A249741(A249725(n)).
a(n) = A249815(A246676(n)).
Other identities. For all n >= 1 the following holds:
a(A000079(n-1)) = A006093(n).

A323885 Sum of A001511 and its Dirichlet inverse.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 08 2019

nonn

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

Cf. A001511, A092673.

Cf. also A318449, A318450, A323365, A323882, A323884, A323887, A323896.

A001511(n) = (1+valuation(n,2));
A092673(n) = (moebius(n)-if(n%2,0,moebius(n/2)));
A323885(n) = (A001511(n)+A092673(n));

from sympy import mobius
def A323885(n): return (n&-n).bit_length()+mobius(n)-(0 if n&1 else mobius(n>>1)) # Chai Wah Wu, Jul 13 2022

a(n) = A001511(n) + A092673(n).

A060572 Tower of Hanoi: the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1 is on move n to move disk A001511 from peg A060571 to peg A060572 (here).

Original entry on oeis.org

1, 2, 2, 1, 0, 1, 1, 2, 2, 0, 0, 2, 1, 2, 2, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 2, 0, 0, 2, 1, 2, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2, 1, 2, 2, 1, 0, 1, 1, 2, 2, 0, 0, 2, 1, 2, 2, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 2, 2, 0, 0, 1, 1, 0, 2, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 2

Offset: 1

Henry Bottomley, Apr 03 2001

If written in a fractal pattern of 4 X 4 squares, skipping the first square, going right then down then right then down, etc.:
   X122 1011 ...
   1011 0200
   2200 1122
   2122 1011
a number of patterns become apparent. Most notably the central diagonal going from the X down and to the right, when the 1's and 2's are reversed, gives the sequence A060571. When the same process is applied to A060571, this sequence emerges. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003

			Start by moving first disk (from peg 0) to peg 1, second disk (from peg 0) to peg 2, first disk (from peg 1) to peg 2, etc., so sequence starts 1,2,2,...

Gary W. Adamson, Comments on A060572
J.-P. Allouche, D. Astoorian, J. Randall, and J. Shallit, Morphisms, squarefree strings, and the Tower of Hanoi puzzle, Amer. Math. Monthly 101 (1994), 651-658.
Index entries for sequences related to Towers of Hanoi

Cf. A001511, A055662, A060571, A060573, A060574, A060575.

a(n) = (- (-1)^valuation(n,2) - n)%3; \\ Kevin Ryde, Aug 07 2021

a(n) = A060571(n) - (-1)^A001511(n) mod 3.
If n > 2^A001511(n) then a(n) = a(n-2^A001511(n)) - (-1)^A001511(n) mod 3, otherwise a(k) = -(-1)^A001511(n) mod 3.
a(n) = A001511(n)-th digit from right of A055662(n).
If a(n)=0 then a(2n)=0, If a(n)=1 then a(2n)=2, If a(n)=2 then a(2n)=1, Thus a(n)=a(4n). - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003
a(5n) = A060571(n) with the 1's and 2s reversed. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003

A094267 First differences of A001511.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 03 2004

For n even, Sum_{k=1..n} a(k) > 0. For n odd, Sum_{k=1..n} a(k) = 0. - James Spahlinger, Oct 13 2013

			G.f. = 1 - x + 2*x^2 - 2*x^3 + x^4 - x^5 + 3*x^6 - 3*x^7 + x^8 - x^9 + ...

James Spahlinger, Table of n, a(n) for n = 0..10000

Absolute values give A050603. Cf. A001511, A005187.

a(n)=(-1)^n*valuation(n+2-n%2,2) \\ Charles R Greathouse IV, Oct 14 2013

{a(n) = my(A); if( n<0, 0, A = sum(k=0, length( binary(n+2)) - 1, x^(2^k) / (1 - x^(2^k)), x^3 * O(x^n));  polcoeff( (A * (1 - x) - x) / x^2, n))}; /* Michael Somos, May 11 2014 */

def A094267(n): return (((m:=n>>1)&~(m+1)).bit_length()+1)*(-1 if n&1 else 1) # Chai Wah Wu, Jul 12 2022

a(n) = (-1)^n * A050603(n).

G.f.: -1/x + (1 - x)*Sum_{k>=0} x^(2^k-2)/(1 - x^(2^k)). - Ilya Gutkovskiy, Feb 28 2017

A115364 a(n) = A000217(A001511(n)), where A001511 is one more than the 2-adic valuation of n, and A000217(n) is the n-th triangular number, binomial(n+1, 2).

Original entry on oeis.org

1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 15, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 21, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 15, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 28, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 15, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1

Offset: 1

Paul Barry, Jan 21 2006

Row sums of A115363. In general, the row sums of ((1,x) - m(x,x^2))^(-2) are obtained by following the ruler function A001511(n) by the solution of the recurrence a(1)=1, a(n) = n*m^(n-1) + a(n-1), n > 1.

The Stephan formula says this is the Dirichlet convolution of A000012 with A104117. - R. J. Mathar, Feb 07 2011

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A000217, A001511, A007814, A104117, A115363.

Cf. also A094290.

Array[PolygonalNumber[IntegerExponent[#, 2] + 1] &, 93] (* Michael De Vlieger, Nov 02 2018 *)

A115364(n) = binomial(valuation(n,2)+2,2); \\ Antti Karttunen, Nov 02 2018

def A115364(n): return (m:=((~n & n-1).bit_length()+1))*(m+1)>>1 # Chai Wah Wu, Jul 02 2022

a(n) = binomial(A007814(n)+2, 2) = binomial(A001511(n)+1, 2).
Dirichlet g.f.: zeta(s)*(2^s/(2^s-1))^2. - Ralf Stephan, Jun 17 2007
Multiplicative with a(2^k) = A000217(k+1), a(p^k) = 1 for odd primes p. - Antti Karttunen, Nov 02 2018
O.g.f.: Sum_{k >= 1} k*x^(2^(k-1))/(1 - x^(2^(k-1))). More generally, if f(n) is an arithmetic function and g(n) := Sum_{k = 1..n} f(k), then Sum_{k >= 1} f(k)*x^(2^(k-1))/(1 - x^(2^(k-1))) = Sum_{n >= 1} g(A001511(n))*x^n. This is the case f(n) = n. - Peter Bala, Mar 26 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Oct 22 2022
More precise asymptotics: Sum_{k=1..n} a(k) ~ 4*n - log(n)*(log(n) + 2*log(4*Pi))/(4*log(2)^2). - Vaclav Kotesovec, Jun 25 2024

Formula corrected and the name changed by Antti Karttunen, Nov 02 2018

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.

A286251 Compound filter: a(n) = P(A001511(1+n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

3, 2, 9, 7, 5, 16, 14, 29, 12, 16, 9, 67, 5, 16, 50, 121, 5, 67, 9, 67, 23, 16, 14, 277, 12, 16, 48, 67, 5, 436, 27, 497, 23, 16, 31, 631, 5, 16, 40, 277, 5, 436, 9, 67, 80, 16, 20, 1129, 12, 67, 31, 67, 5, 277, 40, 277, 23, 16, 9, 1771, 5, 16, 160, 2017, 23, 436, 9, 67, 23, 436, 14, 2557, 5, 16, 94, 67, 23, 436, 20, 1129, 138, 16, 9, 1771, 23, 16, 40, 277, 5

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000027, A001511, A046523, A286161, A286252, A286253, A286254.

A001511(n) = (1+valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286251(n) = (2 + ((A001511(1+n)+A046523(n))^2) - A001511(1+n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286251.txt", n, " ", A286251(n)));

from sympy import factorint
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a001511(n + 1), a046523(n)) # Indranil Ghosh, May 07 2017

(define (A286251 n) (* (/ 1 2) (+ (expt (+ (A001511 (+ 1 n)) (A046523 n)) 2) (- (A001511 (+ 1 n))) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(1+n)+A046523(n))^2) - A001511(1+n) - 3*A046523(n)).

A286260 Compound filter: a(n) = P(A001511(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 1, 39, 4, 8, 1, 157, 79, 47, 4, 39, 22, 8, 4, 600, 37, 782, 11, 256, 1, 47, 4, 157, 466, 233, 11, 39, 106, 47, 1, 2284, 4, 380, 4, 4281, 172, 122, 22, 1132, 211, 8, 56, 256, 742, 47, 4, 600, 1597, 4373, 37, 1278, 352, 122, 37, 157, 11, 1037, 106, 256, 466, 8, 79, 8785, 211, 47, 137, 2083, 4, 47, 37, 19507, 667, 1655, 466, 669, 4, 233, 11, 4661, 7261

Offset: 1

Antti Karttunen, May 07 2017

nonn

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

Cf. A000027, A001511, A161942, A286161, A286162, A286259, A286034.

A001511(n) = (1+valuation(n,2));
A000265(n) = (n >> valuation(n, 2));
A161942(n) = A000265(sigma(n));
A286260(n) = (2 + ((A001511(n)+A161942(n))^2) - A001511(n) - 3*A161942(n))/2;
for(n=1, 16384, write("b286260.txt", n, " ", A286260(n)));

from sympy import factorint, divisors, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a161942(n): return a000265(divisor_sigma(n))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a161942(n)) # Indranil Ghosh, May 07 2017

(define (A286260 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A161942 n)) 2) (- (A001511 n)) (- (* 3 (A161942 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A161942(n))^2) - A001511(n) - 3*A161942(n)).

cp's OEIS Frontend

A092119 EULER transform of A001511.

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A287896 a(n) = A002487(n)*A001511(n).

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A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

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A323885 Sum of A001511 and its Dirichlet inverse.

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A060572 Tower of Hanoi: the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1 is on move n to move disk A001511 from peg A060571 to peg A060572 (here).

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A094267 First differences of A001511.

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A115364 a(n) = A000217(A001511(n)), where A001511 is one more than the 2-adic valuation of n, and A000217(n) is the n-th triangular number, binomial(n+1, 2).

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Mathematica

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