A075300 -id:A075300 - cp's OEIS frontend

A075173 Prime factorization of n encoded by interleaving successive prime exponents in unary to bit-positions given by columns of A075300.

0, 1, 2, 5, 8, 3, 128, 21, 34, 9, 32768, 7, 2147483648, 129, 10, 85, 9223372036854775808, 35, 170141183460469231731687303715884105728, 13, 130, 32769

Offset: 1

Antti Karttunen, Sep 13 2002

nonn

As in A059884, here also we store the exponent e_i of p_i (p1=2, p2=3, p3=5, ...) in the factorization of n to the bit positions given by the column i-1 of A075300 (the exponent of 2 is thus stored to bit positions 0, 2, 4, ..., exponent of 3 to 1, 5, 9, 13, ..., exponent of 5 to 3, 11, 19, 27, 35, ...), but using unary instead of binary system, i.e. we actually store 2^(e_i) - 1 in binary.

This injective mapping from N to N offers an example of the proof given in Cameron's book that any distributive lattice can be represented as a sublattice of the power-set lattice P(X) of some set X. This allows us to implement GCD (A003989) with bitwise AND (A004198) and LCM (A003990) with bitwise OR (A003986). Also, to test whether x divides y, it is enough to check that ((a(x) OR a(y)) XOR a(y)) = A003987(A003986(a(x),a(y)),a(y)) is zero.

			a(24) = 23 because 24 = 2^3 * 3^1 so we add the binary words 10101 and 10 to get 10111 in binary = 23 in decimal and a(25) = 2056 because 25 = 5^2 so we form a binary word 100000001000 = 2056 in decimal.

P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1998, page 191. (12.3. Distributive lattices)

Variant: A075175. Inverse: A075174. Cf. A059884.

A003989(x, y) = A075174(A004198(a(x), a(y))), A003990(x, y) = A075174(A003986(a(x), a(y))).

A075302 Transpose of array A075300.

Original entry on oeis.org

0, 2, 1, 4, 5, 3, 6, 9, 11, 7, 8, 13, 19, 23, 15, 10, 17, 27, 39, 47, 31, 12, 21, 35, 55, 79, 95, 63, 14, 25, 43, 71, 111, 159, 191, 127, 16, 29, 51, 87, 143, 223, 319, 383, 255, 18, 33, 59, 103, 175, 287, 447, 639, 767, 511, 20, 37, 67, 119, 207, 351, 575, 895, 1279

Offset: 0

Antti Karttunen, Sep 12 2002

The top-left corner of the array looks like:
2 4 6 8 10 12 14 ...
5 9 13 17 21 25 29 ...
11 19 27 35 43 51 59 ...
23 39 55 71 87 ...
47 79 ...
This can be used to construct mappings like A059884 and A075173, where the elements of the first row give the bit-positions where to interleave the first element of a finite vector of nonnegative integers, the second row the bit-positions for the second element, etc.

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A075303. Transpose: A075300. The X-projection is given by A025480 and the Y-projection is A007814(n+1). Cf. A002262, A025581.

A075302 := n -> A075300bi(A002262(n), A025581(n));
A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

A075301 Inverse permutation to A075300.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 9, 6, 14, 8, 20, 7, 27, 13, 35, 10, 44, 19, 54, 12, 65, 26, 77, 11, 90, 34, 104, 18, 119, 43, 135, 15, 152, 53, 170, 25, 189, 64, 209, 17, 230, 76, 252, 33, 275, 89, 299, 16, 324, 103, 350, 42, 377, 118, 405, 24, 434, 134, 464, 52, 495, 151, 527, 21

Offset: 0

Antti Karttunen, Sep 12 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A075300. Cf. also A001477, A007814, A025480.

A075301 := n -> A001477bi(A007814(n+1), A025480(n));

A025480 a(2n) = n, a(2n+1) = a(n).

Original entry on oeis.org

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

Offset: 0

David W. Wilson

These are the Grundy values or nim-values for heaps of n beans in the game where you're allowed to take up to half of the beans in a heap. - R. K. Guy, Mar 30 2006. See Levine 2004/2006 for more about this. - N. J. A. Sloane, Aug 14 2016
When n > 0 is written as (2k+1)*2^j then k = a(n-1) and j = A007814(n), so: when n is written as (2k+1)*2^j-1 then k = a(n) and j = A007814(n+1), when n > 1 is written as (2k+1)*2^j+1 then k = a(n-2) and j = A007814(n-1). - Henry Bottomley, Mar 02 2000 [sequence id corrected by Peter Munn, Jun 22 2022]
According to the comment from Deuard Worthen (see Example section), this may be regarded as a triangle where row r=1,2,3,... has length 2^(r-1) and values T(r,2k-1)=T(r-1,k), T(r,2k)=2^(r-1)+k-1; i.e., previous row gives 1st, 3rd, 5th, ... term and 2nd, 4th, ... terms are numbers 2^(r-1),...,2^r-1 (i.e., those following the last one from the previous row). - M. F. Hasler, May 03 2008
Let StB be a Stern-Brocot tree hanging between (pseudo)fractions Left and Right, then StB(1) = mediant(Left,Right) and for n>1: StB(n) = if a(n-1)<>0 and a(n)<>0 then mediant(StB(a(n-1)),StB(a(n))) else if a(n)=0 then mediant(StB(a(n-1)),Right) else mediant(Left,StB(a(n-1))), where mediant(q1,q2) = ((numerator(q1)+numerator(q2)) / (denominator(q1)+denominator(q2))). - Reinhard Zumkeller, Dec 22 2008
This sequence is the unique fixed point of the function (a(0), a(1), a(2), ...) |--> (0, a(0), 1, a(1), 2, a(2), ...) which interleaves the nonnegative integers between the elements of a sequence. - Cale Gibbard (cgibbard(AT)gmail.com), Nov 18 2009
Also the number of remaining survivors in a Josephus problem after the person originally first in line has been eliminated (see A225381). - Marcus Hedbring, May 18 2013
A fractal sequence - see Levine 2004/2006. - N. J. A. Sloane, Aug 14 2016
From David James Sycamore, Apr 29 2020: (Start)
One of a family of fractal sequences, S_k; defined as follows for k >= 2: a(k*n) = n, a(k*n+r) = a((k-1)*n + (r-1)), r = 1..(k-1). S_2 is A025480; S_3 gives: a(3*n) = n, a(3*n + 1) = a(2*n), a(3*n + 2) = a(2*n + 1), which is A263390.
The subsequence of all nonzero terms is A131987. (End)
Similar to but different from A108202. - N. J. A. Sloane, Nov 26 2020
This sequence can be otherwise defined in two alternative (but related) ways, with a(0)=0, as follows: (i) If a(n) is a novel term, then a(n+1) = a(a(n)); if a(n) has been seen before, most recently at a(m), then a(n+1) = n-m (as in A181391). (ii) As above for novel a(n), then if a(n) has been seen before, a(n+1) = smallest k < a(n) which is not already a term. - David James Sycamore, Jul 13 2021
From a binary perspective, the sequence can be seen as even,odd pairs where the odd value is the previous even value, dropping the rightmost bits up to and including the lowest zero bit, aka right-shifted past the lowest clear bit. E.g., (5)101 -> 1, (17)10001 -> (4)100, (29)11101 -> (7)111, (39)100111 -> (2)10. - Joe Nellis, Oct 09 2022

			From Deuard Worthen (deuard(AT)raytheon.com), Jan 27 2006: (Start)
The sequence can be constructed as a triangle as:
  0
  0  1
  0  2  1  3
  0  4  2  5  1  6  3  7
  0  8  4  9  2 10  5 11  1 12  6 13  3 14  7 15
  ...
At each stage we interleave the next 2^m numbers in the previous row. (End)
Left=0/1, Right=1/0: StB=A007305/A047679; Left=0/1, Right=1/1: StB=A007305/A007306; Left=1/3, Right=2/3: StB=A153161/A153162. - _Reinhard Zumkeller_, Dec 22 2008

L. Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Josef Eschgfäller and Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.
Ralf Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.4.
L. Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.
Ralf Stephan, Some divide-and-conquer sequences ....
Ralf Stephan, Table of generating functions.
Paul Tarau, A Groupoid of Isomorphic Data Transformations, Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.
Index entries for sequences related to the Josephus Problem.

The Y-projection of A075300.

Cf. A108202, A138002, A000265, A003602, A103391, A153733, A220466, A225381, A131987, A263390, A181391, A007814.

import Data.List
interleave xs ys = concat . transpose $ [xs,ys]
a025480 = interleave [0..] a025480
-- Cale Gibbard, Nov 18 2009

Cf. comments by Worthen and Hasler.
import Data.List (transpose)
a025480 n k = a025480_tabf !! n !! k
a025480_row n = a025480_tabf !! n
a025480_tabf = iterate (\xs -> concat $
   transpose [xs, [length xs .. 2 * length xs - 1]]) [0]
a025480_list = concat $ a025480_tabf
-- Reinhard Zumkeller, Apr 29 2012

A025480 := proc(n)
    option remember ;
    if type(n,'even') then
        n/2 ;
    else
        procname((n-1)/2) ;
    end if;
end proc:
seq(A025480(n),n=0..100) ; # R. J. Mathar, Jul 16 2020

a[n_] := a[n] = If[OddQ@n, a[(n - 1)/2], n/2]; Table[ a[n], {n, 0, 83}] (* Robert G. Wilson v, Mar 30 2006 *)
Table[BitShiftRight[n, IntegerExponent[n, 2] + 1], {n, 100}] (* IWABUCHI Yu(u)ki, Oct 13 2012 *)

a(n)={while(n%2,n\=2);n\2} \\ M. F. Hasler, May 03 2008

A025480(n)=n>>valuation(n*2+2,2) \\ M. F. Hasler, Apr 12 2012

def A025480(n): return n>>((~(n+1)&n).bit_length()+1) # Chai Wah Wu, Jul 13 2022

A025480 = lambda n: odd_part(n+1)//2
[A025480(n) for n in (0..83)] # Peter Luschny, May 20 2014

a(n) = A003602(n+1) - 1. [Corrected by Max Alekseyev, May 05 2022]
a(n) = (A000265(n+1)-1)/2 = ((n+1)/A006519(n+1)-1)/2.
a(n) = A153733(n)/2. - Reinhard Zumkeller, Dec 31 2008
2^A007814(n+1)*(2*a(n)+1) = n+1. (See functions hd, tl and cons in [Paul Tarau 2009].) - Paul Tarau (paul.tarau(AT)gmail.com), Mar 21 2010
a(3*n + 1) = A173732(n). - Reinhard Zumkeller, Apr 29 2012
a((2*n+1)*2^p-1) = n, p >= 0 and n >= 0. - Johannes W. Meijer, Jan 24 2013
a(n) = n - A225381(n). - Marcus Hedbring, May 18 2013
G.f.: -1/(1-x) + Sum_{k>=0} x^(2^k-1)/(1-2*x^2^(k+1)+x^2^(k+2)). - Ralf Stephan, May 19 2013
a(n) = A049084(A181363(n+1)). - Reinhard Zumkeller, Mar 22 2014
a(n) = floor(n / 2^A001511(n+1)). - Adam Shelly, Mar 05 2019
Recursion: a(0) = 0; a(n + 1) = a(a(n)) if a(n) is a first occurrence of a term, else a(n + 1) = n - a(n-1). - David James Sycamore, Apr 29 2020
a(n) * 2^(A007814(n+1)+1) + 2^A007814(n+1) - 1 = n (equivalent to the formula given in the comment by Paul Tarau). - Ruud H.G. van Tol, Apr 14 2023
Sum_{k=1..n} a(k) = n^2/6 + O(n). - Amiram Eldar, Aug 07 2023

Edited by M. F. Hasler, Mar 16 2018

A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 24, 20, 14, 9, 32, 48, 40, 28, 18, 11, 64, 96, 80, 56, 36, 22, 13, 128, 192, 160, 112, 72, 44, 26, 15, 256, 384, 320, 224, 144, 88, 52, 30, 17, 512, 768, 640, 448, 288, 176, 104, 60, 34, 19, 1024, 1536, 1280, 896, 576, 352, 208, 120

Offset: 0

Henry Bottomley, Apr 12 2000

First column of array is powers of 2, first row is odd numbers, other cells are products of these two, so every positive integer appears exactly once. [Comment edited to match the definition. - L. Edson Jeffery, Jun 05 2015]
An analogous N X N <-> N bijection based, not on the binary, but on the Fibonacci number system, is given by the Wythoff array A035513.
As an array, this sequence (hence also A135764) is the dispersion of the even positive integers. For the definition of dispersion, see the link "Interspersions and Dispersions." The fractal sequence of this dispersion is A003602. - Clark Kimberling, Dec 03 2010

			Northwest corner of array A:
    1     3     5     7     9    11    13    15    17    19
    2     6    10    14    18    22    26    30    34    38
    4    12    20    28    36    44    52    60    68    76
    8    24    40    56    72    88   104   120   136   152
   16    48    80   112   144   176   208   240   272   304
   32    96   160   224   288   352   416   480   544   608
   64   192   320   448   576   704   832   960  1088  1216
  128   384   640   896  1152  1408  1664  1920  2176  2432
  256   768  1280  1792  2304  2816  3328  3840  4352  4864
  512  1536  2560  3584  4608  5632  6656  7680  8704  9728
[Array edited to match the definition. - _L. Edson Jeffery_, Jun 05 2015]
From _Philippe Deléham_, Dec 13 2013: (Start)
a(13-1)=20=2*10, so a(13)=10+A006519(20)=10+4=14.
a(3-1)=3=2*1+1, so a(3)=2^(1+1)=4. (End)
From _Wolfdieter Lang_, Jan 30 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    1
   1:    2    3
   2:    4    6    5
   3:    8   12   10   7
   4:   16   24   20  14   9
   5:   32   48   40  28  18  11
   6:   64   96   80  56  36  22  13
   7:  128  192  160 112  72  44  26  15
   8:  256  384  320 224 144  88  52  30 17
   9:  512  768  640 448 288 176 104  60 34 19
  10: 1024 1536 1280 896 576 352 208 120 68 38 21
  ...
T(3, 2) = 2^1*(2*2+1) = 10. (End)

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Clark Kimberling, Interspersions and Dispersions.
Index entries for sequences that are permutations of the natural numbers

The sequence is a permutation of A000027.
Main diagonal is A014480; inverse permutation is A209268.
Cf. A001511, A002024, A003602, A006519, A025480, A035513, A075300, A135764.

a054582 n k = a054582_tabl !! n !! k
a054582_row n = a054582_tabl !! n
a054582_tabl = iterate
   (\xs@(x:_) -> (2 * x) : zipWith (+) xs (iterate (`div` 2) (2 * x))) [1]
a054582_list = concat a054582_tabl
-- Reinhard Zumkeller, Jan 22 2013

(* Array: *)
Grid[Table[2^m*(2*k + 1), {m, 0, 9}, {k, 0, 9}]] (* L. Edson Jeffery, Jun 05 2015 *)
(* Array antidiagonals flattened: *)
Flatten[Table[2^(m - k)*(2*k + 1), {m, 0, 9}, {k, 0, m}]] (* L. Edson Jeffery, Jun 05 2015 *)

T(m,k)=(2*k+1)<Charles R Greathouse IV, Jun 21 2017

As a sequence, if n is a triangular number, then a(n)=a(n-A002024(n))+2, otherwise a(n)=2*a(n-A002024(n)-1).
a(n) = A075300(n-1)+1.
Recurrence for the sequence: if a(n-1)=2*k is even, then a(n)=k+A006519(2*k); if a(n-1)=2*k+1 is odd, then a(n)=2^(k+1), a(0)=1. - Philippe Deléham, Dec 13 2013
m = A(A001511(m)-1, A003602(m)-1), for each m in A000027. - L. Edson Jeffery, Nov 22 2015
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1), for n >= 0 and k = 0..n. - Wolfdieter Lang, Jan 30 2019

Offset corrected by Reinhard Zumkeller, Jan 22 2013

A037227 If n = 2^mk, k odd, then a(n) = 2m+1.

Original entry on oeis.org

1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 11, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 13, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 11, 1, 3, 1, 5, 1, 3

Offset: 1

N. J. A. Sloane

Take the number of rightmost zeros in the binary expansion of n, double it, and increment it by 1. - Ralf Stephan, Aug 22 2013

Gives the maximum possible number of n X n complex Hermitian matrices with the property that all of their nonzero real linear combinations are nonsingular (see Adams et al. reference). - Nathaniel Johnston, Dec 11 2013

T. D. Noe, Table of n, a(n) for n=1..1024
J. F. Adams, P. D. Lax, and R. S. Phillips, On matrices whose real linear combinations are nonsingular, Proceedings of the American Mathematical Society, 16:318-322, 1965.
D. B. Shapiro, Problem 10456: Anticommuting Matrices, Amer. Math. Monthly, 105 (1998), 565-566.
Index entries for sequences related to binary expansion of n

Cf. A001511, A002487, A007814, A005408, A016825, A017113, A075300, A051062, A220466.

a037227 = (+ 1) . (* 2) . a007814  -- Reinhard Zumkeller, Jun 30 2012

[2*Valuation(n, 2)+1: n in [1..120]]; // Vincenzo Librandi, Jun 19 2019

nmax:=102: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p):= 2*p+1: od: od: seq(a(n), n=1..nmax);  # Johannes W. Meijer, Feb 07 2013

a[n_] := Sum[(-1)^(d+1)*MoebiusMu[d]*DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Dec 31 2012, after Vladeta Jovovic *)
f[n_]:=Module[{z=Last[Split[IntegerDigits[n,2]]]},If[Union[z]={0},2* Length[ z]+1,1]]; Array[f,110] (* Harvey P. Dale, Jun 16 2019, after Ralf Stephan *)
Table[2 IntegerExponent[n, 2] + 1, {n, 120}] (* Vincenzo Librandi, Jun 19 2019 *)

a(n)=2*valuation(n,2)+1 \\ Charles R Greathouse IV, May 21 2015

def A037227(n): return ((~n & n-1).bit_length()<<1)+1 # Chai Wah Wu, Jul 05 2022

maxrow <- 6 # by choice
a <- 1
for(m in 0:maxrow){
for(k in 0:(2^m-1)) {
   a[2^(m+1)    +k] <- a[2^m+k]
   a[2^(m+1)+2^m+k] <- a[2^m+k]
}
   a[2^(m+1)      ] <- a[2^(m+1)] + 2
}
a
# Yosu Yurramendi, May 21 2015

a(n) = Sum_{d divides n} (-1)^(d+1)*mu(d)*tau(n/d). Multiplicative with a(p^e) = 2*e+1 if p = 2; 1 if p > 2. - Vladeta Jovovic, Apr 27 2003
a(n) = a(n-1)+(-1)^n*(a(floor(n/2))+1). - Vladeta Jovovic, Apr 27 2003
a(2*n) = a(n) + 2, a(2*n+1) = 1. a(n) = 2*A007814(n) + 1. - Ralf Stephan, Oct 07 2003
a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012
a((2*n-1)*2^p)  = 2*p+1, p  >= 0 and n >= 1. - Johannes W. Meijer, Feb 07 2013
From Peter Bala, Feb 07 2016: (Start)
a(n) = ( A002487(n-1) + A002487(n+1) )/A002487(n).
a(n*2^(k+1) + 2^k) = 2*k + 1 for n,k >= 0; thus a(2*n+1) = 1, a(4*n+2) = 3, a(8*n+4) = 5, a(16*n+8) = 7 and so on. Note the square array ( n*2^(k+1) + 2^k - 1 )n, k>=0 is the transpose of A075300.
G.f.: Sum_{n >= 0} (2*n + 1)*x^(2^n)/(1 - x^(2^(n+1))). (End)
a(n) = 2*floor(A002487(n-1)/A002487(n))+1 for n > 1. - I. V. Serov, Jun 15 2017
From Amiram Eldar, Nov 29 2022: (Start)
Dirichlet g.f.: zeta(s)*(2^s+1)/(2^s-1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. (End)

More terms from Erich Friedman

A059884 Prime factorization of n encoded by recursively interleaving bits of successive prime exponents.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 128, 5, 32, 9, 32768, 6, 2147483648, 129, 10, 16, 9223372036854775808, 33, 170141183460469231731687303715884105728, 12, 130, 32769

Offset: 1

Marc LeBrun, Feb 06 2001

For n=2^e0*3^e1*5^e2... the alternate (i.e. 2^0,2,4...) bit positions of a(n) give e0, the alternate *remaining* bit positions (i.e. 2^1,5,9...) give e1, the *remaining* alternates (i.e. 2^3,11,19...) give e2 and so on. (Any finite vector of nonnegative integers can be uniquely encoded this way.) Every nonnegative integer appears exactly once in this sequence-despite its outlandish behavior: the next term, a(29) is 2^511 (which has 153 digits), followed by a(30)=11...

Inverse of sequence A059900 considered as a permutation of the nonnegative integers. - Howard A. Landman, Sep 25 2001

			a(360)=a(2^3 * 3^2 * 5^1)=45 thus: ...0 0 0 0 0 0 1 1 -> 3 from 2^3 ...0 0 1 0 -> 2 from 3^2 ...0 1 -> 1 from 5^1 ...00000101101 == 45.

Index entries for sequences that are permutations of the natural numbers

Cf. A075173, A075300, A075302.

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 9, 15, 23, 19, 13, 31, 47, 39, 27, 17, 63, 95, 79, 55, 35, 21, 127, 191, 159, 111, 71, 43, 25, 255, 383, 319, 223, 143, 87, 51, 29, 511, 767, 639, 447, 287, 175, 103, 59, 33, 1023, 1535, 1279, 895, 575, 351, 207, 119, 67, 37

Offset: 1

L. Edson Jeffery, Aug 09 2014

The sequence is a permutation of the odd natural numbers, since A(n,k) = 2*A054582(n-1,k-1) - 1 and A054582 is a permutation of the natural numbers.
For j a natural number, 2*j - 1 appears in row A001511(j) of A.
This is the square array A075300 with the first row omitted. - Peter Bala, Feb 07 2017

			Array begins:
.      1     5     9    13    17     21     25     29     33     37
.      3    11    19    27    35     43     51     59     67     75
.      7    23    39    55    71     87    103    119    135    151
.     15    47    79   111   143    175    207    239    271    303
.     31    95   159   223   287    351    415    479    543    607
.     63   191   319   447   575    703    831    959   1087   1215
.    127   383   639   895  1151   1407   1663   1919   2175   2431
.    255   767  1279  1791  2303   2815   3327   3839   4351   4863
.    511  1535  2559  3583  4607   5631   6655   7679   8703   9727
.   1023  3071  5119  7167  9215  11263  13311  15359  17407  19455

Vincenzo Librandi, Rows n = 0..50, flattened
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A016813, A017101 (rows 1 and 2).
Cf. A000225, A083329, A153894, A086224, A052996, etc. (columns 1-5).
Cf. A005408 (odd natural numbers), A054582.
Cf. A075300.

(* Array: *)
Grid[Table[2^n*(2*k - 1) - 1, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[2^(n - k + 1)*(2*k - 1) - 1, {n, 10}, {k, n}]]

A(n,k) = 2*A054582(n-1,k-1) - 1.

cp's OEIS Frontend

A075173 Prime factorization of n encoded by interleaving successive prime exponents in unary to bit-positions given by columns of A075300.

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A075302 Transpose of array A075300.

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A075301 Inverse permutation to A075300.

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A025480 a(2n) = n, a(2n+1) = a(n).

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A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

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A037227 If n = 2^m*k, k odd, then a(n) = 2*m+1.

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A059884 Prime factorization of n encoded by recursively interleaving bits of successive prime exponents.

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A037227 If n = 2^mk, k odd, then a(n) = 2m+1.

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.