A265341 -id:A265341 - cp's OEIS frontend

A265345 Square array A(row,col): For row=0, A(0,col) = A265341(col), for row > 0, A(row,col) = A265342(A(row-1,col)).

1, 3, 2, 7, 6, 4, 5, 10, 12, 8, 9, 22, 20, 24, 16, 21, 18, 28, 40, 48, 64, 13, 30, 36, 56, 80, 192, 32, 19, 26, 60, 72, 112, 160, 96, 184, 25, 14, 52, 120, 144, 224, 640, 552, 352, 11, 46, 76, 208, 240, 576, 448, 320, 1056, 704, 15, 58, 68, 136, 104, 480, 288, 1720, 1600, 2112, 1408

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), ...
All the terms in the same column are either all divisible by 3, or none of them are.
Reducing A265342 to its constituent sequences gives A265342(n) = A263273(2*A263273(n)). Iterating this function k times starting from n reduces to (because A263273 is an involution, so pairs of them are canceled) to A263273((2^k)*A263273(n)).

			The top left corner of the array:
    1,    3,    7,    5,    9,   21,   13,   19,   25,   11,   15,    39, .
    2,    6,   10,   22,   18,   30,   26,   14,   46,   58,   66,    78, .
    4,   12,   20,   28,   36,   60,   52,   76,   68,   44,   84,   156, .
    8,   24,   40,   56,   72,  120,  208,  136,   88,  232,  168,   624, .
   16,   48,   80,  112,  144,  240,  104,  200,  496,  424,  336,   312, .
   64,  192,  160,  224,  576,  480,  520,  256,  344,  608,  672,  1560, .
   32,   96,  640,  448,  288, 1920, 1144,  512, 1984,  736, 1344,  3432, .
  184,  552,  320, 1720, 1656,  960, 2072, 1024, 1376, 4384, 5160,  6216, .
  352, 1056, 1600,  824, 3168, 4800, 3712, 6040, 5344, 2936, 2472, 11136, .
  ...

Inverse: A265346.
Transpose: A265347.
Leftmost column: A264980.
Topmost row: A265341.
Row index: A265330 (zero-based), A265331 (one-based).
Column index: A265910 (zero-based), A265911 (one-based).
Cf. also A265342.
Related permutations: A263273, A265895.

(define (A265345 n) (A265345bi (A002262 (+ -1 n)) (A025581 (+ -1 n)))) ;; o=1.
(define (A265345bi row col) (A263273 (* (A000079 row) (A263273 (A265341 col))))) ;; Faster than below.
(define (A265345bi row col) (if (= 0 row) (A265341 col) (A265342 (A265345bi (- row 1) col)))) ;; row>=0, col>=0.

For row=0, A(0,col) = A265341(col), for row>0, A(row,col) = A265342(A(row-1,col)).

A(row, col) = A263273((2^row) * A263273(A265341(col))). [The above reduces to this.]

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)).

A270426 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A265341(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 19, 12, 13, 10, 21, 16, 25, 18, 11, 28, 63, 38, 23, 24, 17, 26, 27, 20, 15, 42, 37, 32, 57, 50, 73, 36, 31, 22, 39, 56, 33, 126, 91, 76, 117, 46, 61, 48, 43, 34, 75, 52, 79, 54, 29, 40, 67, 30, 49, 84, 157, 74, 45, 64, 147, 114, 41, 100, 183, 146, 77, 72, 35, 62, 51, 44, 55, 78, 53, 112, 193, 66

Offset: 1

Antti Karttunen, Mar 17 2016

This sequence can be represented as a binary tree. Each left hand child will contain 2n, and each right hand child will contain A265341(n), when the parent node contains n:
                                    1
                 ................../ \..................
                2                                       3
      4......../ \........7                   6......../ \........5
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  8       9          14       19         12       13         10       21
16 25   18 11      28  63   38  23     24  17   26  27     20  15   42  37
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A270425.

Cf. A265341.

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A265341(a(n)).

A265353 Permutation of nonnegative integers: a(n) = A264985(A263273(n)).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 6, 9, 12, 5, 7, 19, 8, 13, 31, 24, 28, 37, 15, 11, 33, 18, 27, 30, 21, 36, 39, 14, 16, 46, 23, 25, 73, 69, 55, 64, 17, 22, 58, 26, 40, 94, 78, 85, 112, 51, 34, 100, 72, 82, 91, 75, 109, 118, 42, 32, 96, 20, 35, 105, 60, 99, 102, 45, 29, 87, 54, 81, 84, 57, 90, 93, 48, 38, 114, 63, 108, 111, 66, 117, 120, 41

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its odd bisection.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265354.
Cf. A263273, A264985.
Cf. also A265341, A265351, A265355, A265356.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a264985(a263273(n)) # Indranil Ghosh, May 22 2017

(define (A265353 n) (A264985 (A263273 n)))

a(n) = A264985(A263273(n)).

A265342 Permutation of even numbers: a(n) = 2 * A265351(n).

Original entry on oeis.org

0, 2, 4, 6, 8, 22, 12, 10, 16, 18, 20, 58, 24, 26, 76, 66, 64, 70, 36, 14, 40, 30, 28, 34, 48, 46, 52, 54, 56, 166, 60, 62, 184, 174, 172, 178, 72, 74, 220, 78, 80, 238, 228, 226, 232, 198, 68, 202, 192, 190, 196, 210, 208, 214, 108, 38, 112, 42, 44, 130, 120, 118, 124, 90, 32, 94, 84, 82, 88, 102, 100, 106, 144

Offset: 0

Antti Karttunen, Dec 07 2015

nonn

Iterating this sequence as 1, a(1), a(a(1)), a(a(a(1))), ... yields A264980.

Antti Karttunen, Table of n, a(n) for n = 0..9841

Cf. A265351.

Cf. also A265341, A263273, A264980.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a263272(n): return a263273(2*n)/2
def a(n): return 2*a263272(a263273(n)) # Indranil Ghosh, May 25 2017

Scheme
```
(define (A265342 n) (* 2 (A265351 n)))
```

a(n) = 2 * A265351(n).

cp's OEIS Frontend

A265345 Square array A(row,col): For row=0, A(0,col) = A265341(col), for row > 0, A(row,col) = A265342(A(row-1,col)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A270426 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A265341(a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A265353 Permutation of nonnegative integers: a(n) = A264985(A263273(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

A265342 Permutation of even numbers: a(n) = 2 * A265351(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula