A135764 -id:A135764 - cp's OEIS frontend

A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 11, 14, 20, 24, 16, 13, 22, 28, 40, 48, 32, 17, 26, 44, 56, 80, 96, 64, 19, 34, 52, 88, 112, 160, 192, 128, 9, 38, 68, 104, 176, 224, 320, 384, 256, 23, 18, 76, 136, 208, 352, 448, 640, 768, 512, 29, 46, 36, 152, 272, 416, 704, 896, 1280, 1536, 1024, 15, 58, 92, 72, 304, 544, 832, 1408, 1792, 2560, 3072, 2048, 31, 30

Offset: 1

Antti Karttunen, Jan 24 2015

Shares with A135764 and A253551 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling.

			The top left corner of the array:
   1,  3,  5,   7,  11,  13,  17,  19,   9,  23,  29,  15,  31,  37,  41,  43,
   2,  6, 10,  14,  22,  26,  34,  38,  18,  46,  58,  30,  62,  74,  82,  86,
   4, 12, 20,  28,  44,  52,  68,  76,  36,  92, 116,  60, 124, 148, 164, 172,
   8, 24, 40,  56,  88, 104, 136, 152,  72, 184, 232, 120, 248, 296, 328, 344,
  16, 48, 80, 112, 176, 208, 272, 304, 144, 368, 464, 240, 496, 592, 656, 688,
...

Inverse: A254054.
Similar or related permutations: A135764, A253551, A064216, A254051.
Cf. A000079, A001511, A002260, A004736, A064989, A135765, A249745, A254049, A254050.

A(row,col) = A135764(row, A249745(col)). [Is otherwise the same array as A135764, but the column positions have been permuted by A249745.]
A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = 2^(row-1) * A254050(col). [The above expands to this.]
a(n) = A064989(A135765(n)).
As a composition of other permutations:
a(n) = A064216(A254051(n)). [As an array: A(row,col) = A064216(A254051(row,col)).]

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).

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).

A253551 Square array: A(row,col) = 2^(row-1) * 1+(2*A156552(col)) = A156552(A246278(row,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, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 11, 18, 28, 40, 48, 32, 17, 22, 36, 56, 80, 96, 64, 15, 34, 44, 72, 112, 160, 192, 128, 13, 30, 68, 88, 144, 224, 320, 384, 256, 19, 26, 60, 136, 176, 288, 448, 640, 768, 512, 33, 38, 52, 120, 272, 352, 576, 896, 1280, 1536, 1024, 23, 66, 76, 104, 240, 544, 704, 1152, 1792, 2560, 3072, 2048, 65, 46

Offset: 1

Antti Karttunen, Jan 03 2015

Shares with A135764 the property that A001511(n) = k for all terms n on row k and when going downwards in each column, terms grow by doubling.

			The top left corner of the array:
   1,  3,  5,   7,   9, 11,  17,  15,  13,  19,  33,  23,  65,  35,  21,
   2,  6, 10,  14,  18, 22,  34,  30,  26,  38,  66,  46, 130,  70,  42,
   4, 12, 20,  28,  36, 44,  68,  60,  52,  76, 132,  92, 260, 140,  84,
   8, 24, 40,  56,  72, 88, 136, 120, 104, 152, 264, 184, 520, 280, 168,
  16, 48, 80, 112, 144,176, 272, 240, 208, 304, 528, 368,1040, 560, 336,
...

Inverse: A253552.
Cf. A000079, A001511, A002260, A005408, A004736, A005940, A005941, A156552, A246278.
Differs from A135764 for the first time at n=22, where a(22) = 17, while A135764(22) = 13.

A(row,col) = A156552(A246278(row,col)).
A(row,col) = A135764(row, A005941(col)). [Is otherwise the same array as A135764, but the column positions have been permuted by A005941.]
A(row,col) = 2^(row-1) * ((2*A005941(col)) - 1) = 2^(row-1) * A005408(A156552(col)). [The above expands to this.]
As a composition of other permutations:
a(n) = A156552(A246278(n+1)). [When all three sequences are interpreted as one-dimensional sequences.]

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)).

A254055 Square array: A(row,col) = A003602(A254051(row,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, 2, 1, 1, 1, 3, 2, 6, 12, 4, 4, 9, 1, 9, 21, 5, 3, 13, 48, 102, 31, 3, 7, 30, 75, 36, 10, 183, 2, 15, 39, 6, 112, 426, 912, 274, 7, 18, 22, 58, 264, 669, 160, 684, 1641, 8, 10, 7, 129, 345, 198, 1003, 3828, 8202, 2461, 1, 6, 57, 156, 193, 517, 2370, 6015, 2871, 3076, 14763, 5, 24, 66, 85, 117, 1155, 3099, 889, 9022, 34446, 73812, 22144

Offset: 1

Antti Karttunen, Jan 27 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,A(row+1,col)).

What the resulting odd number will be, is given by A254101(row+1,col).

			The top left corner of the array:
    1,   2,    1,    2,    4,    5,     3,     2,    7,    8,     1, ...
    1,   1,    6,    9,    3,    7,    15,    18,   10,    6,    24, ...
    3,  12,    1,   13,   30,   39,    22,     7,   57,   66,    18, ...
    4,   9,   48,   75,    6,   58,   129,   156,   85,   25,   210, ...
   21, 102,   36,  112,  264,  345,   193,   117,  507,  588,    79, ...
   31,  10,  426,  669,  198,  517,  1155,  1398,  760,  441,  1884, ...
  183, 912,  160, 1003, 2370, 3099,  1732,    66, 4557, 5286,  1413, ...
  274, 684, 3828, 6015,  889, 4648, 10389, 12576, 6835,  496, 16950, ...
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Cf. A003602, A002260, A004736.

Related arrays: A135764, A135765, A254051, A254101, A254102.

A254101 Square array A(row,col) = A000265(A254051(row,col)).

Original entry on oeis.org

1, 3, 1, 1, 1, 5, 3, 11, 23, 7, 7, 17, 1, 17, 41, 9, 5, 25, 95, 203, 61, 5, 13, 59, 149, 71, 19, 365, 3, 29, 77, 11, 223, 851, 1823, 547, 13, 35, 43, 115, 527, 1337, 319, 1367, 3281, 15, 19, 13, 257, 689, 395, 2005, 7655, 16403, 4921, 1, 11, 113, 311, 385, 1033, 4739, 12029, 5741, 6151, 29525

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 at the first row of array A135764 as x_new = A135764(1,A254055(row+1,col)) and it is given here as A(row+1,col) = A000265(A254051(row+1,col)).

That number's column index in array A135765 is then given by A254102(row+1,col).

			The top left corner of the array:
    1,    3,    1,     3,    7,    9,     5,     3,    13,    15,     1, ...
    1,    1,   11,    17,    5,   13,    29,    35,    19,    11,    47, ...
    5,   23,    1,    25,   59,   77,    43,    13,   113,   131,    35, ...
    7,   17,   95,   149,   11,  115,   257,   311,   169,    49,   419, ...
   41,  203,   71,   223,  527,  689,   385,   233,  1013,  1175,   157, ...
   61,   19,  851,  1337,  395, 1033,  2309,  2795,  1519,   881,  3767, ...
  365, 1823,  319,  2005, 4739, 6197,  3463,   131,  9113, 10571,  2825, ...
  547, 1367, 7655, 12029, 1777, 9295, 20777, 25151, 13669,   991, 33899, ...
etc.

Cf. A000265, A002260, A004736, A003961, A139391, A249745.

Related arrays: A135764, A135765, A254051, A254055, A254102.

(define (A254101 n) (A254101bi (A002260 n) (A004736 n)))
(define (A254101bi row col) (+ -1 (* 2 (A254055bi row col))))
;; Alternative definition:
(define (A254101v2 n) (A254101biv2 (A002260 n) (A004736 n)))
(define (A254101biv2 row col) (A003961 (A254055bi row (A249745 col))))

A(row,col) = A000265(A254051(row,col)).
A(row,col) = (2*A254055(row,col))-1.
A(row,col) = A003961(A254055(row, A249745(col))).
A(row+1,col) = A139391(A135765(row,col)).
As compositions of one-dimensional sequences:
a(n) = A000265(A254051(n)).
a(n) = (2*A254055(n))-1.

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)).

A349102 Increase the odd part of n to the next greater odd number.

Original entry on oeis.org

3, 6, 5, 12, 7, 10, 9, 24, 11, 14, 13, 20, 15, 18, 17, 48, 19, 22, 21, 28, 23, 26, 25, 40, 27, 30, 29, 36, 31, 34, 33, 96, 35, 38, 37, 44, 39, 42, 41, 56, 43, 46, 45, 52, 47, 50, 49, 80, 51, 54, 53, 60, 55, 58, 57, 72, 59, 62, 61, 68, 63, 66, 65, 192, 67, 70, 69

Offset: 1

Kevin Ryde, Mar 26 2022

This is +2 at the bit position of the odd part of n, that being the least significant 1-bit.
The least significant run of 1-bits changes from 0111..111 in n to 1000..001 in a(n).
Arrays A054582 and A135764 arrange terms into rows having the same number of trailing 0 bits. a(n) is the term to the right of n, i.e., next in its row.

			n    = 3448 = binary 1101 0111 1 000
a(n) = 3464 = binary 1101 1000 1 000

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000265 (odd part), A171977 (2 at odd part), A285326.

Arrays: A054582, A135764.

Array[# + 2^(IntegerExponent[#, 2] + 1) &, 67] (* Michael De Vlieger, Mar 27 2022 *)

PARI
```
a(n) = n + 2<
				
```

a(n) = n + A171977(n).
a(2*n) = 2*a(n).
a(2*n+1) = 2*n + 3.

A135766 Multiply sequence A007775 (1 7 11 13 ...) by sequence A000351 (1 5 25 125 ...).

Original entry on oeis.org

1, 7, 5, 11, 35, 25, 13, 55, 175, 125, 17, 65, 275, 875, 625, 19, 85, 325, 1375, 4375, 3125, 23, 95, 425, 1625, 6875, 21875, 15625, 29, 115, 475, 2125, 8125, 34375, 109375, 78125, 31, 145, 575, 2375, 10625, 40625, 171875, 546875, 390625, 37, 155, 725, 2875

Offset: 1

Alford Arnold, Nov 29 2007

			A(13) = 275 since A007775(3) * A000351(3) = 11 times 25.

Cf. A135764, A135765, A007775 (not divisible by 2, 3, or 5), A000351 (powers of five).

T(n,k) = A007775(n-k)*A000351(k), n>0, 0<=kR. J. Mathar, Jan 07 2008

More terms from R. J. Mathar, Jan 07 2008

cp's OEIS Frontend

A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,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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A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

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

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A253551 Square array: A(row,col) = 2^(row-1) * 1+(2*A156552(col)) = A156552(A246278(row,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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A254102 Square array A(row,col) = A253887(A254055(row,col)) = A126760(A254101(row,col)).

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A254055 Square array: A(row,col) = A003602(A254051(row,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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A254101 Square array A(row,col) = A000265(A254051(row,col)).

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

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A349102 Increase the odd part of n to the next greater odd number.

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