A254053 -id:A254053 - cp's OEIS frontend

A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 11, 18, 28, 40, 48, 32, 13, 22, 36, 56, 80, 96, 64, 15, 26, 44, 72, 112, 160, 192, 128, 17, 30, 52, 88, 144, 224, 320, 384, 256, 19, 34, 60, 104, 176, 288, 448, 640, 768, 512, 21, 38, 68, 120, 208, 352, 576, 896, 1280, 1536, 1024, 23, 42, 76, 136, 240, 416, 704, 1152, 1792, 2560, 3072, 2048, 25, 46, 84, 152, 272, 480, 832, 1408, 2304, 3584, 5120, 6144, 4096, 27, 50, 92, 168, 304, 544, 960, 1664, 2816

Offset: 1

Alford Arnold, Nov 29 2007

The array in A135764 is identical to the array in A054582 [up to the transposition and different indexing. - Clark Kimberling, Dec 03 2010; comment amended by Antti Karttunen, Feb 03 2015; please see the illustration in Example section].
The array gives a bijection between the natural numbers N and N^2. A more usual bijection is to take the natural numbers A000027 and write them in the usual OEIS square array format. However this bijection has the advantage that it can be formed by iterating the usual bijection between N and 2N. - Joshua Zucker, Nov 04 2011
The array can be used to determine the configurations of k-th Towers of Hanoi moves, by labeling odd row terms C,B,A,C,B,A,... and even row terms B,C,A,B,C,A,.... Then given k equal to or greater than term "a" in each n-th row, but less than the next row term, record the label A, B, or C for term "a". This denotes the peg position for the disc corresponding to the n-th row. For example, with k = 25, five discs are in motion since the binary for 25 = 11001, five bits. We find that 25 in row 5 is greater than 16 labeled C, but less than 48. Thus, disc 5 is on peg C. In the 4th row, 25 is greater than 24 (a C), but less than 40, so goes onto the C peg. Similarly, disc 3 is on A, 2 is on A, and disc 1 is on A. Thus, discs 2 and 3 are on peg A, while 1, 4, and 5 are on peg C. - Gary W. Adamson, Jun 22 2012
Shares with arrays A253551 and A254053 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling. - Antti Karttunen, Feb 03 2015
Let P be the infinite palindromic word having initial word 0 and midword sequence (1,2,3,4,...) = A000027. Row n of the array A135764 gives the positions of n-1 in S. ("Infinite palindromic word" is defined at A260390.) - Clark Kimberling, Aug 13 2015
The probability distribution series 1 = 2/3 + 4/15 + 16/255 + 256/65535 + ... + A001146(n-1)/A051179(n) governs the proportions of terms in A001511 from row n of the array. In A001511(1..15) there are ((2/3) * 15) = ten terms from row one of the array, ((4/15) * 15) = four terms from row two, and ((16/255) * 15) = one (rounded), giving one term from row three (a 4). - Gary W. Adamson, Dec 16 2021
From Gary W. Adamson, Dec 30 2021: (Start)
Subarrays representing the number of divisors of an integer can be mapped on the table. For 60, write the odd divisors on the top row: 1, 3, 5, 15. Since 60 has 12 divisors, let the left column equal 1, 2, 4, where 4 is the highest power of 2 dividing 60. Multiplying top row terms by left column terms, we get the result:
  1   3   5  15
  2   6  10  30
  4  12  20  60. (End)

			The table begins
   1,  3,   5,   7,   9,  11,  13,  15,  17,  19,  21,  23, ...
   2,  6,  10,  14,  18,  22,  26,  30,  34,  38,  42,  46, ...
   4, 12,  20,  28,  36,  44,  52,  60,  68,  76,  84,  92, ...
   8, 24,  40,  56,  72,  88, 104, 120, 136, 152, 168, 184, ...
  16, 48,  80, 112, 144, 176, 208, 240, 272, 304, 336, 368, ...
  32, 96, 160, 224, 288, 352, 416, 480, 544, 608, 672, 736, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (i.e., 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000079(3-1) * A005408(1-1) = 2^2 * 1 = 4.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3 (row,col) in above table) and A(3,3) = A000079(3-1) * A005408(3-1) = 2^2 * 5 = 20.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000079(2-1) * A005408(6-1) = 2^1 * 11 = 22.

Transpose: A054582.
Inverse permutation: A249725.
Column 1: A000079.
Row 1: A005408.
Cf. A001511 (row index), A003602 (column index, both one-based).
Related arrays: A135765, A253551, A254053, A254055.
Cf. A000027, A002260, A004736, A064989, A135766, A249746.
Cf. also permutations A246675, A246676, A249741, A249811, A249812.
Cf. A260390.
Cf. A001146, A051179.

seq(seq(2^(j-1)*(2*(i-j)+1),j=1..i),i=1..20); # Robert Israel, Feb 03 2015

f[n_] := Block[{i, j}, {1}~Join~Flatten@ Last@ Reap@ For[j = 1, j <= n, For[i = j, i > 0, Sow[2^(j - i - 1)*(2 i + 1)], i--], j++]]; f@ 10 (* Michael De Vlieger, Feb 03 2015 *)

a(n) = {s = ceil((1 + sqrt(1 + 8*n)) / 2); r = n - binomial(s-1, 2) - 1;k = s - r - 2; 2^r * (2 * k + 1) } \\ David A. Corneth, Feb 05 2015

(define (A135764 n) (A135764bi (A002260 n) (A004736 n)))
(define (A135764bi row col) (* (A000079 (- row 1)) (+ -1 col col)))
;; Antti Karttunen, Feb 03 2015

From Antti Karttunen, Feb 03 2015: (Start)
A(row, col) = 2^(row-1) * ((2*col)-1) = A000079(row-1) * A005408(col-1).
A(row,col) = A064989(A135765(row,A249746(col))).
A(row,col) = A(row+1,col)/2 [discarding the topmost row and halving the rest of terms gives the array back].
A(row,col) = A(row,col+1) - A000079(row) [discarding the leftmost column and subtracting 2^{row number} from the rest of terms gives the array back].
(End)
G.f.: ((2*x+1)*Sum_{i>=0} 2^i*x^(i*(i+1)/2) + 2*(1-2*x)*Sum_{i>=0} i*x^(i*(i+1)/2) + (1-6*x)*Sum_{i>=0} x^(i*(i+1)/2) - 1 - 2*x)*x/(1-2*x)^2. These sums are related to Jacobi theta functions. - Robert Israel, Feb 03 2015

More terms from Sean A. Irvine, Nov 23 2010

Name amended and the illustration of array in the example section transposed by Antti Karttunen, Feb 03 2015

A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

Original entry on oeis.org

1, 3, 4, 6, 13, 7, 9, 18, 10, 12, 28, 15, 25, 63, 16, 19, 33, 39, 21, 43, 22, 24, 88, 27, 61, 48, 30, 46, 58, 31, 34, 138, 60, 36, 73, 37, 40, 123, 72, 42, 313, 45, 67, 78, 49, 94, 93, 81, 51, 163, 52, 54, 193, 55, 57, 103, 64, 102, 213, 105, 85, 108, 172, 66, 118, 69, 127, 438, 70, 75, 133, 111, 109, 303

Offset: 1

Antti Karttunen, Jan 24 2015

nonn

Shift the prime factorization of odd numbers one step towards larger primes, add one and divide by two.

			For n = 8, the eighth odd number is 2*8 - 1 = 15 = 3*5 = prime(2) * prime(3). By adding one to both prime indices, we get prime(3) * prime(4) = 5*7 = 35, and (35+1)/2 = 18, thus a(8) = 18. Here prime(n) = A000040(n).

Cf. A032766 (omitting the initial 0, the same sequence sorted into ascending order).
Also a permutation of A253888.
Cf. A000040, A003961, A048673, A249735, A249746, A254050, A254051, A254053.

a(n) = A048673(2*n-1) = (1+A003961(2*n-1)) / 2 = (1+A249735(n)) / 2.

a(n) = A032766(A249746(n)).

A191450 Dispersion of (3*n-1), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 14, 23, 11, 6, 41, 68, 32, 17, 7, 122, 203, 95, 50, 20, 9, 365, 608, 284, 149, 59, 26, 10, 1094, 1823, 851, 446, 176, 77, 29, 12, 3281, 5468, 2552, 1337, 527, 230, 86, 35, 13, 9842, 16403, 7655, 4010, 1580, 689, 257, 104, 38, 15, 29525

Offset: 1

Clark Kimberling, Jun 05 2011

Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A016789(n-1), t(n) = A032766(n) [from term A032766(1) onward] and u(n) = A253887(n). [Author's original comment edited by Antti Karttunen, Jan 24 2015]

For other examples of such sequences, please see the Crossrefs section.

			The northwest corner of the square array:
  1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...
  3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...
  4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...
  6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...
  7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...
  9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...
  etc.
The leftmost column is A032766, and each successive column to the right of it is obtained by multiplying the left neighbor on that row by three and subtracting one, thus the second column is (3*1)-1, (3*3)-1, (3*4)-1, (3*6)-1, (3*7)-1, (3*9)-1, ... = 2, 8, 11, 17, 20, 26, ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254047.
Transpose: A254051.
Column 1: A032766.
Cf. A007051, A057198, A199109, A199113 (rows 1-4).
Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).
Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.

A191450 := proc(r, c)
    option remember;
    if c = 1 then
        A032766(r) ;
    else
        A016789(procname(r, c-1)-1) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n-1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191450 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

a(n,k)=3^(n-1)*(k*3\2*2-1)\2+1 \\ =3^(n-1)*(k*3\2-1/2)+1/2, but 30% faster. - M. F. Hasler, Jan 20 2015

(define (A191450 n) (A191450bi (A002260 n) (A004736 n)))
(define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))
(define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on L. Edson Jeffery's direct formula.
;; Antti Karttunen, Jan 21 2015

Conjecture: A(n,k) = (3 + (2*A032766(n) - 1)*A000244(k))/6. - L. Edson Jeffery, with slight changes by Antti Karttunen, Jan 21 2015

a(n) = A254051(A038722(n)). [When both this and transposed array A254051 are interpreted as one-dimensional sequences.] - Antti Karttunen, Jan 22 2015

Example corrected and description clarified by Antti Karttunen, Jan 24 2015

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset: 1

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

			The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Inverse: A254052.
Transpose: A191450.
Row 1: A032766.
Cf. A007051, A057198, A199109, A199113 (columns 1-4).
Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
Array A135765(n,k) = 2*A(n,k) - 1.
Other related arrays: A254055, A254101, A254102.
Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015
A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
A(n+1,k) = A165355(A135765(n,k)).
As a composition of related permutations. All sequences interpreted as one-dimensional:
a(n) = A048673(A254053(n)). [Proved above.]
a(n) = A191450(A038722(n)). [Transpose of array A191450.]

A135765 Distribute the odd numbers in columns based on the occurrence of "3" in each prime factorization; square array A(row, col) = 3^(row-1) * A007310(col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 5, 3, 7, 15, 9, 11, 21, 45, 27, 13, 33, 63, 135, 81, 17, 39, 99, 189, 405, 243, 19, 51, 117, 297, 567, 1215, 729, 23, 57, 153, 351, 891, 1701, 3645, 2187, 25, 69, 171, 459, 1053, 2673, 5103, 10935, 6561, 29, 75, 207, 513, 1377, 3159, 8019, 15309, 32805

Offset: 1

Alford Arnold, Nov 28 2007

The Table can be constructed by multiplying sequence A000244 by A007310.
From Antti Karttunen, Jan 26 2015: (Start)
A permutation of odd numbers. Adding one to each term and then dividing by two gives a related table A254051, which for any odd number, located in this array as x = A(row,col), gives the result at A254051(row+1,col) after one combined Collatz step (3x+1)/2 -> x (A165355) has been applied.
Each odd number n occurs here in position A(A007949(n), A126760(n)).
Compare also to A135764.
(End)

			The top left corner of the array:
    1,    5,    7,   11,   13,   17,   19,   23,   25,   29,   31,   35, ...
    3,   15,   21,   33,   39,   51,   57,   69,   75,   87,   93,  105, ...
    9,   45,   63,   99,  117,  153,  171,  207,  225,  261,  279,  315, ...
   27,  135,  189,  297,  351,  459,  513,  621,  675,  783,  837,  945, ...
   81,  405,  567,  891, 1053, 1377, 1539, 1863, 2025, 2349, 2511, 2835, ...
  243, 1215, 1701, 2673, 3159, 4131, 4617, 5589, 6075, 7047, 7533, 8505, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (that is 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000244(3-1) * A007310(1) = 3^2 * 1 = 9.
For n = 9, we have [A002260(9), A004736(9)] = [3, 2] (9 corresponds to location 3,2) and A(3,2) = A000244(3-1) * A007310(2) = 3^2 * 5 = 9*5 = 45.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3) and A(3,3) = A000244(3-1) * A007310(3) = 3^2 * 7 = 9*7 = 63.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000244(2-1) * A007310(6) = 3^1 * 17 = 51.

Row 1: A007310.
Column 1: A000244.
Cf. A007949 (row index), A126760 (column index).
Cf. also A000265, A002260, A003961, A004736, A005408, A165355, A249745.
Related arrays: A135764, A254051, A254055, A254101, A254102.

N:= 20:
B:= [seq(op([6*n+1,6*n+5]),n=0..floor((N-1)/2))]:
[seq(seq(3^j*B[i-j],j=0..i-1),i=1..N)]; # Robert Israel, Jan 26 2015

From Antti Karttunen, Jan 26 2015: (Start)
With both row and col starting from 1:
A(row, col) = A000244(row-1) * A007310(col) = 3^(row-1) * A007310(col).
a(n) = (2*A254051(n))-1.
a(n) = A003961(A254053(n)).
Above in array form:
A(row,col) = A003961(A254053(row,col)) = A003961(A135764(row,A249745(col))).
(End)

Name amended and examples edited by Antti Karttunen, Jan 26 2015

A253888 a(0) = 1; for n >= 1: a(n) = A048673(1+(2*A064216(n))).

Original entry on oeis.org

1, 3, 4, 6, 7, 13, 18, 15, 9, 63, 39, 28, 43, 12, 10, 27, 31, 16, 19, 138, 88, 123, 45, 25, 78, 48, 30, 81, 24, 73, 55, 105, 22, 36, 108, 72, 438, 111, 21, 37, 303, 33, 148, 42, 93, 87, 103, 213, 54, 91, 58, 298, 171, 34, 363, 165, 172, 198, 102, 49, 69, 163, 76, 46, 115, 228, 333, 288, 61, 135, 319, 90, 130, 75, 52

Offset: 0

Antti Karttunen, Jan 22 2015

nonn

When A048673 is represented as a binary tree, then the node k which contains value n = A048673(k) has as its right child a(n) = A048673(2k+1).

Antti Karttunen, Table of n, a(n) for n = 0..8192

Same sequence sorted into ascending order: A032766.
Also a permutation of A254049.
Cf. A048673, A064216, A253889, A253890, A254051, A254053.

a(0) = 1; for n >= 1: a(n) = A048673(1+(2*A064216(n))).

Also, for n >= 1: a(n) = A254049(1+A064216(n)).

A254050 Permutation of odd numbers: a(n) = (2*(A249745(n))) - 1 = A064989(A007310(n)).

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 9, 23, 29, 15, 31, 37, 41, 43, 25, 47, 21, 53, 59, 33, 61, 67, 71, 35, 73, 79, 39, 83, 55, 51, 89, 97, 101, 103, 107, 109, 57, 65, 49, 27, 113, 127, 85, 131, 137, 77, 69, 139, 149, 87, 151, 95, 157, 163, 121, 167, 45, 173, 179, 93, 91, 181, 191, 193, 197, 115, 111, 119, 199, 123

Offset: 1

Antti Karttunen, Jan 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A007310, A064989, A249745, A254049, A254051, A254053.

a(n) = (2*(A249745(n))) - 1.

a(n) = A064989(A007310(n)).

A254054 Permutation of natural numbers: a(n) = A254052(A048673(n)).

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 7, 10, 37, 8, 11, 9, 16, 12, 67, 15, 22, 47, 29, 13, 172, 17, 46, 14, 137, 23, 862, 18, 56, 80, 79, 21, 232, 30, 326, 58, 92, 38, 407, 19, 106, 192, 121, 24, 1712, 57, 154, 20, 821, 155, 497, 31, 191, 905, 466, 25, 742, 68, 211, 94, 254, 93, 4187, 28, 781, 255, 277, 39, 1177, 353, 301, 70, 352, 107, 3322, 48, 1129, 437, 379, 26

Offset: 1

Antti Karttunen, Jan 24 2015

nonn

This is an inverse permutation to A254053, see comments there.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A254053.

Similar or related permutations: A048673, A254052.

(define (A254054 n) (A254052 (A048673 n)))

a(n) = A254052(A048673(n)).

A265895 Square array: A(row,col) = A263273(A265345(row,col)) = 2^row * A263273(A265341(col)).

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 15, 18, 28, 40, 48, 32, 13, 30, 36, 56, 80, 96, 64, 11, 26, 60, 72, 112, 160, 192, 128, 17, 22, 52, 120, 144, 224, 320, 384, 256, 19, 34, 44, 104, 240, 288, 448, 640, 768, 512, 21, 38, 68, 88, 208, 480, 576, 896, 1280, 1536, 1024, 39, 42, 76, 136, 176, 416, 960, 1152, 1792, 2560, 3072, 2048

Offset: 1

Antti Karttunen, Dec 18 2015

Square array A(row,col) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...

Shares with arrays A135764, A253551 and A254053 the property that odd terms are on the top row and when going downward in each column, terms grow by doubling.

			The top left corner of the array:
    1,   3,    5,    7,    9,   15,   13,   11,   17,   19,   21,   39,
    2,   6,   10,   14,   18,   30,   26,   22,   34,   38,   42,   78,
    4,  12,   20,   28,   36,   60,   52,   44,   68,   76,   84,  156,
    8,  24,   40,   56,   72,  120,  104,   88,  136,  152,  168,  312,
   16,  48,   80,  112,  144,  240,  208,  176,  272,  304,  336,  624,
   32,  96,  160,  224,  288,  480,  416,  352,  544,  608,  672, 1248,
   64, 192,  320,  448,  576,  960,  832,  704, 1088, 1216, 1344, 2496,
  128, 384,  640,  896, 1152, 1920, 1664, 1408, 2176, 2432, 2688, 4992,
  256, 768, 1280, 1792, 2304, 3840, 3328, 2816, 4352, 4864, 5376, 9984,
...

Inverse permutation: A265896.
The top row: 1+(2*A263273(n)).
Cf. A263273, A265341, A265345.
Differs from A135764 for the first time at n=16, where a(16) = 15, while A135764(16) = 11.

A(row,col) = A263273(A265345(row,col)).

A(row,col) = 2^row * A263273(A265341(col)).

cp's OEIS Frontend

A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

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A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

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A191450 Dispersion of (3*n-1), read by antidiagonals.

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A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n*(2*floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A135765 Distribute the odd numbers in columns based on the occurrence of "3" in each prime factorization; square array A(row, col) = 3^(row-1) * A007310(col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A253888 a(0) = 1; for n >= 1: a(n) = A048673(1+(2*A064216(n))).

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A254050 Permutation of odd numbers: a(n) = (2*(A249745(n))) - 1 = A064989(A007310(n)).

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A254054 Permutation of natural numbers: a(n) = A254052(A048673(n)).

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A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...