A000079 -id:A000079 - cp's OEIS frontend

A209229 Characteristic function of powers of 2, cf. A000079.

0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Mar 06 2012

Essentially the same as A036987 (the Fredholm-Rueppel sequence).
Completely multiplicative with a(2^e) = 1, a(p^e) = 0 for odd primes p. - Mitch Harris, Apr 19 2005
Moebius transform of A001511. - R. J. Mathar, Jun 20 2014

			x + x^2 + x^4 + x^8 + x^16 + x^32 + x^64 + x^128 + x^256 + x^512 + x^1024 + ...

Michel Dekking, Michel Mendes France and Alf van der Poorten, "Folds", The Mathematical Intelligencer, Vol. 4, No. 3 (1982), pp. 130-138 & front cover, and Vol. 4, No. 4 (1982), pp. 173-181 (printed in two parts).
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Antti Karttunen, Table of n, a(n) for n = 0..65537 (terms 0..1000 from G. C. Greubel)
Index entries for characteristic functions
Index to divisibility sequences

Cf. A001511, A029837 (partial sums), A087003 (moebius transform), A090678, A104977, A154955 (Dirichlet inverse).

Cf. A069517, A154269, A201219, A255738.

int a (unsigned long n) { return n & !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */

a209229 n | n < 2 = n
          | n > 1 = if m > 0 then 0 else a209229 n'
          where (n',m) = divMod n 2

A209229 := proc(n)
    if n <= 0 then
        0 ;
    elif n = 1 then
        1;
    elif type (n,'odd') or A001221(n) > 1 then
        0 ;
    else
        1;
    end if;
end proc:
seq(A209229(n),n=0..40) ; # R. J. Mathar, Jan 07 2021

a[n_] := Boole[n == 2^IntegerExponent[n, 2]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 06 2014 *)
Table[If[IntegerQ[Log[2,n]],1,0],{n,0,100}] (* Harvey P. Dale, Jun 24 2018 *)

a(n)=n==1<Charles R Greathouse IV, Mar 07 2012

{a(n) = if( n<2 || n%2, n==1, isprimepower(n) > 0)} \\ Michael Somos, Jan 03 2013

def A209229(n): return int(not(n&-n)^n) if n else 0 # Chai Wah Wu, Jul 08 2022

a(A000079(n)) = 1; a(A057716(n)) = 0.
a(n+1) = A036987(n).
a(n) = if n < 2 then n else (if n is even then a(n/2) else 0).
The generating function g(x) satisfies g(x) - g(x^2) = x. - Joerg Arndt, May 11 2010
Dirichlet g.f.: 1/(1 - 2^(-s)). - R. J. Mathar, Mar 07 2012
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x / (1 + x / (1 - x / ...)))))) = x / (1 + b(1) * x / (1 + b(2) * x / (1 + b(3) * x / ...))) where b(n) = (-1)^ A090678(n+1). - Michael Somos, Jan 03 2013
With a(0) = 0 removed is convolution inverse of A104977. - Michael Somos, Jan 03 2013
From Antti Karttunen, Nov 19 2017: (Start)
a(n) = abs(A154269(n)).
For n > 1, a(n) = A069517(n)/2 = 2 - A201219(n). (End)
a(n) = A048298(n)/n. - R. J. Mathar, Jan 07 2021
a(n) = floor((2^n)/n) - floor((2^n - 1)/n), for n>=1. - Ridouane Oudra, Oct 15 2021

A131577 Zero followed by powers of 2 (cf. A000079).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Paul Curtz, Aug 29 2007, Dec 06 2007

A000079 is the main entry for this sequence.
Binomial transform of A000035.
Essentially the same as A034008 and A000079.
a(n) = a(n-1)-th even natural numbers (A005846) for n > 1. - Jaroslav Krizek, Apr 25 2009
Where record values greater than 1 occur in A083662: A000045(n)=A083662(a(n)). - Reinhard Zumkeller, Sep 26 2009
Number of compositions of natural number n into parts >0.
The signed sequence 0, 1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, ... is the Lucas U(-2,0) sequence. - R. J. Mathar, Jan 08 2013
In computer programming, these are the only unsigned numbers such that k&(k-1)=0, where & is the bitwise AND operator and numbers are expressed in binary. - Stanislav Sykora, Nov 29 2013
Also the 0-additive sequence: a(n) is the smallest number larger than a(n-1) which is not the sum of any subset of earlier terms, with initial values {0, 1, 2}. - Robert G. Wilson v, Jul 12 2014
Also the smallest nonnegative superincreasing sequence: each term is larger than the sum of all preceding terms. Indeed, an equivalent definition is a(0)=0, a(n+1)=1+sum_{k=0..n} a(k). - M. F. Hasler, Jan 13 2015

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adi Dani, Compositions of natural numbers over arithmetic progressions
Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.
J. T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, 18 (2015), #15.10.7.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for Lucas sequences

Cf. A000079, A003945, A042950, A020406, A046045, A011782.

int is (unsigned long n) { return !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */

a131577 = (`div` 2) . a000079
a131577_list = 0 : a000079_list  -- Reinhard Zumkeller, Dec 09 2012

[(2^n-0^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 10 2011

A131577 := proc(n)
    if n =0 then
        0;
    else
        2^(n-1) ;
    end if;
end proc: # R. J. Mathar, Jul 22 2012

Floor[2^Range[-1, 33]] (* Robert G. Wilson v, Sep 02 2007 *)
Join[{0}, 2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

a(n)=1<Charles R Greathouse IV, Sep 15 2012

def A131577(n): return 1<Chai Wah Wu, Sep 09 2023

a(n) = floor(2^(n-1)). - Robert G. Wilson v, Sep 02 2007
G.f.: x/(1-2*x); a(n) = (2^n-0^n)/2. - Paul Barry, Jan 05 2009
E.g.f.: exp(x)*sinh(x). - Geoffrey Critzer, Oct 28 2012
E.g.f.: x/T(0) where T(k) = 4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Mar 17 2013
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n, 2*k-1). - Taras Goy, Jan 02 2025

More terms from Robert G. Wilson v, Sep 02 2007
Edited by N. J. A. Sloane, Sep 13 2007
Edited by M. F. Hasler, Jan 13 2015

A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 18, 21, 256, 23, 12, 25, 14, 27, 512, 29, 1024, 31, 26, 33, 20, 35, 2048, 37, 42, 39, 4096, 41, 8192, 43, 22, 45, 16384, 47, 24, 49, 50, 51, 32768, 53, 36, 55, 66, 57, 65536, 59, 131072, 61, 38, 63, 52, 65, 262144, 67, 74, 69

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

Consider the square array A246278, and also A246275 which is obtained from the former when one is subtracted from each term.
In A246278 the even numbers occur at the top row, and all the rows below that contain only odd numbers, those subsequent terms in each column having been obtained by shifting all primes present in the prime factorization of number immediately above to one larger indices with A003961.
To compute a(n): we do the same process in reverse, by shifting primes in the prime factorization of n+1 step by step to smaller primes, until after k >= 0 such shifts with A064989, the result is even, with the smallest prime present being 2.
We subtract one from this even number and shift the binary expansion of the resulting odd number k positions left (i.e. multiply it with 2^k), which will be the result of a(n).
In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.
A055396(n+1) tells on which row of A246275 n is, which is equal to the row of A246278 on which n+1 is.
A246277(n+1) tells in which column of A246275 n is, which is equal to the column of A246278 in which n+1 is.

			Consider 54 = 55-1. To find 55's position in array A246278, we start shifting its prime factorization 55 = 5 * 11 = p_3 * p_5, step by step: p_2 * p_4 (= 3 * 7 = 21), until we get an even number: p_1 * p_3 = 2*5 = 10.
This tells us that 55 is on row 3 and column 5 (= 10/2) of array A246278, thus 54 occurs in the same position at array A246275. In array A135764 the same position contains number (2^(3-1)) * (10-1) = 4*9 = 36, thus a(54) = 36.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246676.
More recursed variants: A246677, A246683.
Even bisection halved: A246679.
Other related permutations: A054582, A135764, A246274, A246275, A246276.
Cf. A000079, A003961, A005408, A055396, A064989, A246277, A246278.
a(n) differs from A156552(n+1) for the first time at n=13, where a(13) = 14, while A156552(14) = 17.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246675(n) = { my(k=0); n++; while((n%2), n = A064989(n); k++); n--; while(k>0, n = 2*n; k--); n; };
for(n=1, 2048, write("b246675.txt", n, " ", A246675(n)));

(define (A246675 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (-1+ (* 2 (A246277 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1)  * ((2*A246277(n+1))-1).
As a composition of related permutations:
a(n) = A135764(A246276(n)).
a(n) = A054582(A246274(n)-1).
Other identities. For all n >= 0:
a(A005408(n)) = A005408(n). [Fixes the odd numbers.]

A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079).

Original entry on oeis.org

1, 5, 31, 209, 1471, 10625, 78079, 580865, 4361215, 32978945, 250806271, 1916280833, 14698053631, 113104519169, 872801042431, 6751535300609, 52337071357951, 406468580343809, 3162019821780991, 24634626678980609, 192179216026959871

Offset: 0

Joseph Abate, Jun 15 2010

Hankel transform is A133460.

			a(3) = (1,4,10,20)dot(1,2,4,8) = 209.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian motion and generalized Catalan numbers, Journal of Integer Sequences, Vol. 14 (2011).
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.

Row sums of A091811.

Cf. A383326, A383716.

a := n -> binomial(2*n+2,n+1)*hypergeom([-n, n + 1], [n + 2], -1)/2:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 21 2017

CoefficientList[Series[(3-Sqrt[1-8*x])/(2*(1+x)*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
Table[Sum[2^k*Binomial[n + k, k], {k, 0, n}], {n, 0, 20}] (* Michael De Vlieger, Oct 28 2016 *)
a[n_] := (-1)^(n + 1) - 2^(n + 1) (2n + 1) Binomial[2n, n] Hypergeometric2F1[1, 2n + 2, n + 2, 2]/(n + 1); Array[a, 22, 0] (* Robert G. Wilson v, Jul 21 2018 *)

a(n) = Sum_{k = 0..n} A046899(n,k)*2^k = Sum_{k = 0..n} 2^k * binomial(n+k,k).
G.f.: (1/3)*(4/sqrt(1 - 8*x) - 1/(1 - x*c(2*x))) with c(x) the g.f. of the Catalan numbers A000108.
a(n) = (1/3)*(4*2^n*A000984(n) - A064062(n)).
a(n) + a(n+1) = 6*2^n*A001700(n).
O.g.f.: (3 - sqrt(1 - 8*x))/(2*(1 + x)*sqrt(1 - 8*x)). - Peter Bala, Apr 10 2012
a(n) = 2^n *binomial(2+2*n,1+n)*2F1(1, 2+2*n; 2+n;-1). - Olivier Gérard, Aug 19 2012
D-finite with recurrence n*a(n) = (7*n - 4)*a(n-1) + 4*(2*n - 1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(3n+2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = Sum_{k = 0..n} binomial(k+n,n) * binomial(2*n+1,n-k). - Vladimir Kruchinin, Oct 28 2016
a(n) = 1/2*(n + 1)*binomial(2*n+2,n+1)*Sum_{k = 0..n} binomial(n,k)/(n + k + 1). - Peter Bala, Feb 21 2017
a(n) = binomial(2*n+2,n+1)*hypergeom([-n, n+1], [n+2], -1)/2. - Peter Luschny, Feb 21 2017
a(n) = (-1)^(n+1) - 2^(n+1)*(2*n+1)*binomial(2*n,n)*hypergeom([1, 2*n+2], [n+2], 2)/(n+1). - John M. Campbell, Jul 14 2018
From Akiva Weinberger, Dec 06 2024: (Start)
a(n) = (2*n + 1)!/(n!^2) * Integral_{t=0..1} (t + t^2)^n dt.
a(n) = (Integral_{t=0..1} (t + t^2)^n dt) / (Integral_{t=0..1} (t - t^2)^n dt). (End)
a(n) = (-1)^n * Sum_{k=0..n} binomial(2*n+1,k) * (-2)^k. - André M. Timpanaro, Dec 15 2024
From Seiichi Manyama, Aug 04 2025: (Start)
a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(n+1)). (End)

A246683 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 128, 31, 18, 29, 256, 63, 12, 25, 14, 19, 512, 21, 1024, 127, 26, 33, 20, 255, 2048, 61, 58, 35, 4096, 57, 8192, 511, 30, 125, 16384, 23, 24, 49, 50, 27, 32768, 37, 36, 1023, 66, 41, 65536, 2047, 131072, 253, 62, 51, 52, 65, 262144, 39, 122, 509, 524288, 4095, 1048576, 121

Offset: 1

Antti Karttunen, Sep 06 2014

nonn

See the comments in A246675. This is otherwise similar permutation, except that after having reached an even number 2m when we have shifted the prime factorization of n+1 k steps towards smaller primes, here, in contrary to A246675, we don't shift the binary representation of the odd number 2m-1, but instead of an odd number (2*a(m))-1 the same number (k) of bit-positions leftward, i.e. multiply it with 2^k.

See also the comments at the inverse permutation A246684.

			Consider 44 = 45-1. To find 45's position in array A246278, we start shifting its prime factorization 45 = 3 * 3 * 5 = p_2 * p_2 * p_3, step by step, until we get an even number, which in this case happens immediately after the first step, as p_1 * p_1 * p_2 = 2*2*3 = 12. 12 is in the 6th column of A246278, thus we take [here a(6) is computed recursively in the same way:] (2*a(6))-1 = (2*8)-1 = 15, "1111" in binary, and shift it one bit left (that is, multiply by 2), to give 2*15 = 30, thus a(44) = 30.

Antti Karttunen, Table of n, a(n) for n = 1..2049
Index entries for sequences that are permutations of the natural numbers

Inverse: A246684.
Variants: A246675, A246677.
Cf. A000051, A000079, A005408, A055396, A246277, A246278,
A249813, A250244.
Differs from A249813 for the first time at n=20, where a(20) = 18, while A249813(20) = 14.

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).
As a composition of other permutations:
a(n) = A249813(A250244(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back].
For all n >= 0, the following holds: a(A000051(n)) = A000051(n). [Numbers of the form 2^n + 1 are among the fixed points].

A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 128, 31, 14, 29, 256, 63, 12, 25, 18, 19, 512, 21, 1024, 127, 30, 33, 20, 255, 2048, 61, 26, 27, 4096, 57, 8192, 511, 22, 125, 16384, 23, 24, 49, 34, 35, 32768, 37, 28, 1023, 62, 41, 65536, 2047, 131072, 253, 58, 59, 36, 65, 262144, 39, 126, 509, 524288, 4095, 1048576, 121, 50, 51, 40, 53

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

This sequence is a "recursed variant" of A249812.

See also the comments at the inverse permutation A249814.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A249814.
Similar or related permutations: A246683, A249812, A250243.
Cf. A000079, A006093, A055396, A078898.
Differs from A246683 for the first time at n=20, where a(20) = 14, while A246683(20) = 18.

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).
As a composition of other permutations:
a(n) = A246683(A250243(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A006093(n)) = A000079(n-1).

A082900 a(n) = A082894(n)-A000079(n), the difference of 2^n and the number closest to it and divisible by n.

Original entry on oeis.org

0, 0, 1, 0, -2, 2, -2, 0, 1, -4, -2, -4, -2, -4, 7, 0, -2, 8, -2, 4, -8, -4, -2, 8, -7, -4, 1, 12, -2, -4, -2, 0, -8, -4, 17, 8, -2, -4, -8, -16, -2, 20, -2, -16, -17, -4, -2, -16, 19, -24, -8, -16, -2, 26, 12, 24, -8, -4, -2, -16, -2, -4, -8, 0, -32, 2, -2, -16, -8, 26, -2, 8, -2, -4, 7, -16, -18, 14, -2, -16, 1, -4, -2, 20, -32, -4, -8

Offset: 1

Labos Elemer, Apr 22 2003

n=5:2^5=32 and number divisible by n=5 and closest to 32 is 30 = A082894(5), a(5)=30-32=-2 is the corresponding difference.

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

Cf. A082893, A082894, A082895, A082896, A082898, A082899, A082901.

Table[n*Floor[(Floor[n/2]+2^n)/n]-2^n, {n, 1, 100}]

A082894(n) = (n*(((n\2)+2^n)\n));
A082900(n) = (A082894(n)-(2^n)); \\ Antti Karttunen, Feb 18 2023

A082900(n) = { my(x=2^n); for(k=0,oo,if(!((x-k)%n),return(-k), if(!((x+k)%n),return(k)))); }; \\ Antti Karttunen, Feb 18 2023

def A082900(n): return (m:=(k:=1<>1))-m%n-k # Chai Wah Wu, Apr 23 2025

a(n)=n*floor[(floor(n/2)+2^n)/n]-2^n

A116623 a(0)=1, a(2n) = a(n)+A000079(A000523(2n)), a(2n+1) = 3*a(n) + A000079(A000523(2n+1)+1).

Original entry on oeis.org

1, 5, 7, 19, 11, 29, 23, 65, 19, 49, 37, 103, 31, 85, 73, 211, 35, 89, 65, 179, 53, 143, 119, 341, 47, 125, 101, 287, 89, 251, 227, 665, 67, 169, 121, 331, 97, 259, 211, 601, 85, 223, 175, 493, 151, 421, 373, 1087, 79, 205, 157, 439, 133, 367, 319, 925, 121

Offset: 0

Antti Karttunen, Feb 20 2006. Proposed by Pierre Lamothe (plamothe(AT)aei.ca), May 21 2004

Viewed as a binary tree, this is (1); 5; 7,19; 11,29,23,65; ... Related to the parity vectors of Collatz and Terras trajectories.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. a(n) = A116640(A059893(n)). a(A000225(n)) = A001047(n+1). For n>= 1 a(A000079(n)) = A062709(n+1). A116641 gives the terms in ascending order and without duplicates.

A116623 := proc(n)
    option remember;
    if n = 0 then
        1;
    elif type(n,'even') then
        procname(n/2)+2^A000523(n) ;
    else
        3*procname(floor(n/2))+2^(1+A000523(n)) ;
    end if;
end proc: # R. J. Mathar, Nov 28 2016

a[n_] := a[n] = Which[n == 0, 1, EvenQ[n], a[n/2] + 2^Floor@Log2[n], True, 3a[Floor[n/2]] + 2^(1 + Floor@Log2[n])];
Table[a[n], {n, 0, 56}] (* Jean-François Alcover, Sep 01 2023 *)

A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2A246277(2n+1))-1), a(2n+1) = 1 + 2a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 26, 129, 20, 27, 2048, 257, 42, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 50, 131, 32768, 29, 36, 71, 66, 1025, 65536, 43, 131072, 2049, 38, 63, 52, 53, 262144, 259, 74, 41

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

See the comments in A246675. This is otherwise similar permutation, except for odd numbers, which are here recursively permuted by the emerging permutation itself. The even bisection halved gives A246679, the odd bisection from a(3) onward with one subtracted and then halved gives this sequence back.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246678. Variants: A246675, A246683.
Even bisection halved: A246679.
Cf. A000079, A055396, A246277.
a(n) differs from A156552(n+1) for the first time at n=32, where a(32) = 26, while A156552(33) = 34.

a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2*A246277(2n+1))-1), a(2n+1) = 1 + 2*a(n).

A249812 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 14, 21, 256, 23, 12, 25, 18, 27, 512, 29, 1024, 31, 22, 33, 20, 35, 2048, 37, 26, 39, 4096, 41, 8192, 43, 30, 45, 16384, 47, 24, 49, 34, 51, 32768, 53, 28, 55, 38, 57, 65536, 59, 131072, 61, 42, 63, 36, 65, 262144, 67, 46, 69, 524288, 71, 1048576, 73, 50, 75, 40, 77, 2097152, 79, 54, 81, 4194304, 83, 44

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A249741, the sieve of Eratosthenes minus 1. 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 A054582 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A249811.
Cf. A000079, A005408, A006093, A055396, A078898.
Similar or related permutations: A249813 ("deep variant"), A246675, A249816, A054582, A114881, A250252, A135764, A249741, A249742.
Differs from A246675 for the first time at n=20, where a(20)=14, while A246675(20)=18.

(define (A249812 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (+ -1 (* 2 (A078898 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).
As a composition of related permutations:
a(n) = A054582(A250252(n)-1).
a(n) = A135764(A249742(n)).
a(n) = A246675(A249816(n)).
Other identities. For all n >= 1 the following holds:
a(A006093(n)) = A000079(n-1).

cp's OEIS Frontend

A209229 Characteristic function of powers of 2, cf. A000079.

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A131577 Zero followed by powers of 2 (cf. A000079).

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A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

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A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079).

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A246683 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).

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A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).

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A082900 a(n) = A082894(n)-A000079(n), the difference of 2^n and the number closest to it and divisible by n.

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A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2A246277(2n+1))-1), a(2n+1) = 1 + 2a(n).