A254051 -id:A254051 - cp's OEIS frontend

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), ...

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

A254102 Square array A(row,col) = A253887(A254055(row,col)) = A126760(A254101(row,col)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 8, 3, 3, 6, 1, 6, 14, 1, 2, 9, 32, 68, 21, 2, 5, 20, 50, 24, 7, 122, 1, 10, 26, 4, 75, 284, 608, 183, 5, 12, 15, 39, 176, 446, 107, 456, 1094, 2, 7, 5, 86, 230, 132, 669, 2552, 5468, 1641, 1, 4, 38, 104, 129, 345, 1580, 4010, 1914, 2051, 9842

Offset: 1

Antti Karttunen, Jan 28 2015

Starting with an odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 is found in A254051(row+1,col), and after iterated [i.e., we divide all powers of 2 out] Collatz step: x_new <- A139391(x) = A000265(3x+1) the resulting odd number x_new is located A135764(1,A254055(row+1,col)).
What the resulting odd number will be, is given by A254101(row+1,col) =  A000265(A254051(row+1,col)).
That number's column index in array A135765 is then given by A(row+1,col).

			The top left corner of the array:
     1,    1,    1,    1,     3,     1,     2,    1,     5,     2,     1,
     1,    1,    4,    6,     2,     5,    10,   12,     7,     4,    16,
     2,    8,    1,    9,    20,    26,    15,    5,    38,    44,    12,
     3,    6,   32,   50,     4,    39,    86,  104,    57,    17,   140,
    14,   68,   24,   75,   176,   230,   129,   78,   338,   392,    53,
    21,    7,  284,  446,   132,   345,   770,  932,   507,   294,  1256,
   122,  608,  107,  669,  1580,  2066,  1155,   44,  3038,  3524,   942,
   183,  456, 2552, 4010,   593,  3099,  6926, 8384,  4557,   331, 11300,
  1094, 5468, 1914, 6015, 14216, 18590, 10389, 6288, 27338, 31712,   530,
etc.

Cf. A000265, A126760, A253887, A254048.

Related arrays: A135764, A135765, A254051, A254055, A254101.

(define (A254102 n) (A254102bi (A002260 n) (A004736 n)))
;; In turn using either one of these three bivariate functions:
(define (A254102 n) (A254102bi (A002260 n) (A004736 n)))
(define (A254102bi row col) (A126760 (A254051bi row col)))
(define (A254102bi row col) (A253887 (A254055bi row col)))
(define (A254102bi row col) (A126760 (A254101bi row col)))

A(row,col) = A126760(A254051(row,col)) = A126760(A254101(row,col)).
A(row,col) = A253887(A254055(row,col)).
A(row+1,col) = A254048(A135765(row,col)).

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)).

A257943 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (1 + 3^(n-1)(2k - 1))/2, n,k >= 1.

Original entry on oeis.org

1, 2, 2, 5, 5, 3, 14, 14, 8, 4, 41, 41, 23, 11, 5, 122, 122, 68, 32, 14, 6, 365, 365, 203, 95, 41, 17, 7, 1094, 1094, 608, 284, 122, 50, 20, 8, 3281, 3281, 1823, 851, 365, 149, 59, 23, 9, 9842, 9842, 5468, 2552, 1094, 446, 176, 68, 26, 10

Offset: 1

L. Edson Jeffery, May 13 2015

			Array A begins:
.     1      2      3      4      5       6       7       8       9      10
.     2      5      8     11     14      17      20      23      26      29
.     5     14     23     32     41      50      59      68      77      86
.    14     41     68     95    122     149     176     203     230     257
.    41    122    203    284    365     446     527     608     689     770
.   122    365    608    851   1094    1337    1580    1823    2066    2309
.   365   1094   1823   2552   3281    4010    4739    5468    6197    6926
.  1094   3281   5468   7655   9842   12029   14216   16403   18590   20777
.  3281   9842  16403  22964  29525   36086   42647   49208   55769   62330
.  9842  29525  49208  68891  88574  108257  127940  147623  167306  186989

Cf. A191450, A254051.

(* Array: *)
Grid[Table[(1 + 3^(n - 1)*(2*k - 1))/2, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[(1 + 3^(n - k)*(2*k - 1))/2, {n, 10}, {k, n}]]

A279724 Transpose of array A257943.

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 4, 8, 14, 14, 5, 11, 23, 41, 41, 6, 14, 32, 68, 122, 122, 7, 17, 41, 95, 203, 365, 365, 8, 20, 50, 122, 284, 608, 1094, 1094, 9, 23, 59, 149, 365, 851, 1823, 3281, 3281, 10, 26, 68, 176, 446, 1094, 2552, 5468, 9842, 9842

Offset: 1

L. Edson Jeffery, Dec 17 2016

			Array A begins as:
.      1   2   5   14   41   122   365   1094   3281    9842
.      2   5  14   41  122   365  1094   3281   9842   29525
.      3   8  23   68  203   608  1823   5468  16403   49208
.      4  11  32   95  284   851  2552   7655  22964   68891
.      5  14  41  122  365  1094  3281   9842  29525   88574
.      6  17  50  149  446  1337  4010  12029  36086  108257
.      7  20  59  176  527  1580  4739  14216  42647  127940
.      8  23  68  203  608  1823  5468  16403  49208  147623
.      9  26  77  230  689  2066  6197  18590  55769  167306
.     10  29  86  257  770  2309  6926  20777  62330  186989

Cf. A191450, A254051, A257943.

(* Array: *)
Grid[Table[(1 + 3^(k - 1)*(2*n - 1))/2, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[(1 + 3^(k - 1)*(2*(n - k) + 1))/2, {n, 10}, {k, n}]]

A(n,k) = (1 + 3^(k-1)*(2*n - 1))/2, n,k >= 1.

cp's OEIS Frontend

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), ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A254102 Square array A(row,col) = A253887(A254055(row,col)) = A126760(A254101(row,col)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Links

Crossrefs

Formula

A257943 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (1 + 3^(n-1)*(2*k - 1))/2, n,k >= 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A279724 Transpose of array A257943.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A257943 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (1 + 3^(n-1)(2k - 1))/2, n,k >= 1.