A269387 -id:A269387 - cp's OEIS frontend

A269379 a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).

1, 3, 5, 9, 7, 15, 11, 21, 19, 27, 13, 33, 17, 39, 35, 45, 23, 51, 31, 57, 49, 63, 25, 69, 29, 75, 65, 81, 37, 87, 55, 93, 79, 99, 59, 105, 41, 111, 95, 117, 43, 123, 47, 129, 109, 135, 53, 141, 85, 147, 125, 153, 61, 159, 73, 165, 139, 171, 103, 177, 67, 183, 155, 189, 113, 195, 71, 201, 169, 207, 77, 213, 101, 219, 185, 225, 83

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

a(n) = the number located immediately below n in A255127 (square array generated by Ludic sieve) in the same column where n itself is, or in other words, the number removed in the next filtering stage at the same step as when n was removed in the A260738(n)-th stage.

Permutation of odd numbers.

Antti Karttunen, Table of n, a(n) for n = 1..32999

Cf. A255127, A260738, A260739.
Cf. A269171, A269356, A269358, A269382, A269385, A269387 (sequences that use this function).
Cf. A269380 (left inverse).
Cf. also A250469, A269369.

(define (A269379 n) (if (= 1 n) n (A255127bi (+ (A260738 n) 1) (A260739 n)))) ;; Code for A255127bi given in A255127.

a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).
Other identities. For all n >= 1:
A269380(a(n)) = n.

A269171 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269379(a(A268674(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 21, 22, 25, 24, 19, 26, 27, 28, 29, 30, 37, 32, 33, 34, 35, 36, 41, 46, 39, 40, 43, 42, 47, 44, 45, 50, 53, 48, 31, 38, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 71, 74, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 89, 82, 75, 92, 59, 78, 91, 80, 81, 86, 97, 84, 79, 94, 87, 88, 107, 90, 85, 100

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

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

Inverse: A269172.
Cf. A000035, A003309, A008578, A055396, A078898, A268674, A255127, A269379.
Related or similar permutations: A260741, A260742, A269355, A269357, A255421, A252754, A252756, A269385, A269387.
Cf. also A269393 (a(3n)/3) and A269395.
Differs from A255407 for the first time at n=38, where a(38) = 46, while A255407(38) = 38.

a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, a(n) = A269379(a(A268674(n))).
a(1) = 1, for n > 1, a(n) = A255127(A055396(n), a(A078898(n))).
As a composition of other permutations:
a(n) = A269385(A252756(n)).
a(n) = A269387(A252754(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [Preserves the parity of n.]
a(A008578(n)) = A003309(n). [Maps noncomposites to Ludic numbers.]

A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269380(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 12, 19, 14, 33, 128, 23, 256, 65, 18, 35, 512, 21, 24, 31, 22, 129, 20, 27, 1024, 25, 34, 39, 2048, 29, 4096, 67, 30, 257, 8192, 47, 28, 513, 26, 131, 16384, 37, 48, 71, 38, 1025, 40, 43, 32768, 49, 66, 63, 36, 45, 65536, 259, 46, 41, 131072, 55, 96, 2049, 130, 51, 262144, 69, 44

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269387.
Cf. A269380.
Related permutations: A260742, A269386, A269172.
Cf. also A252754, A269378.
Differs from A156552, A252754 and A246677(n-1) for the first time at n=19, which here a(19)=12, instead of 128.

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269380(n)).
As a composition of other permutations:
a(n) = A252754(A269172(n)).
a(n) = A269378(A260742(n)).

A302025 Permutation of natural numbers mapping ordinary factorization to "Ludic factorization": a(1) = 1, a(2n) = 2*a(n), a(A003961(n)) = A269379(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 27, 22, 25, 24, 19, 26, 21, 28, 29, 30, 37, 32, 39, 34, 35, 36, 41, 46, 63, 40, 43, 54, 47, 44, 33, 50, 53, 48, 31, 38, 75, 52, 61, 42, 65, 56, 99, 58, 67, 60, 71, 74, 57, 64, 95, 78, 77, 68, 135, 70, 83, 72, 89, 82, 51, 92, 59, 126, 91, 80, 45, 86, 97, 108, 155, 94, 147, 88

Offset: 1

Antti Karttunen, Apr 03 2018

nonn

See comments and examples in A302032 to see how Ludic factorization proceeds.

Cf. A302026 (inverse permutation).
Cf. A156552, A250245, A269171, A269387 (similar or related permutations).
Cf. A003961, A064989, A269379.

\\ With A269379 precomputed:
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A302025(n) = if(1==n,n,if(!(n%2),2*A302025(n/2),A269379(A302025(A064989(n)))));
(Scheme, with memoization-macro definec)
(definec (A302025 n) (cond ((= 1 n) n) ((even? n) (* 2 (A302025 (/ n 2)))) (else (A269379 (A302025 (A064989 n))))))

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269379(a(A064989(2n+1))).
a(n) = A269171(A250245(n)).
a(n) = A269387(A156552(n)).

A269385 Tree of Ludic sieve, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A269379(a(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 6, 5, 16, 21, 18, 19, 12, 15, 10, 7, 32, 45, 42, 49, 36, 51, 38, 31, 24, 33, 30, 35, 20, 27, 14, 11, 64, 93, 90, 109, 84, 123, 98, 85, 72, 105, 102, 125, 76, 111, 62, 55, 48, 69, 66, 79, 60, 87, 70, 59, 40, 57, 54, 65, 28, 39, 22, 13, 128, 189, 186, 229, 180, 267, 218, 191, 168, 249, 246, 305, 196, 291, 170, 151, 144

Offset: 0

Antti Karttunen, Mar 01 2016

Permutation of natural numbers obtained from the Ludic sieve. Note the indexing: Domain starts from 0, range from 1.
This sequence can be represented as a binary tree. Each left hand child is obtained by doubling the parent's contents, and each right hand child is obtained by applying A269379 to the parent's contents:
                                     1
                                     |
                  ...................2...................
                 4                                       3
       8......../ \........9                   6......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       21         18       19         12       15         10       7
32  45   42  49     36  51   38  31     24  33   30  35     20  27   14 11
etc.
Sequence A269387 is obtained from the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A269386.
Cf. A000035, A269379.
Cf. A003309 (right edge of the tree).
Related or similar permutations: A163511, A260741, A269387, A269171.
Cf. also A252755, A269375.

a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A269379(a(n)).
As a composition of related permutations:
a(n) = A269171(A252755(n)).
a(n) = A260741(A269375(n)).
Other identities. For all n >= 2:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n from a(2)=4 onward.]

A269377 Tree of Lucky sieve: a(0) = 1, a(1) = 2; after which a(2n) = A269369(a(n)), a(2n+1) = 2*a(n).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 11, 12, 19, 10, 17, 16, 13, 18, 35, 28, 39, 22, 29, 24, 27, 38, 23, 20, 61, 34, 41, 32, 15, 26, 47, 36, 123, 70, 77, 56, 57, 78, 59, 44, 103, 58, 65, 48, 45, 54, 107, 76, 81, 46, 53, 40, 91, 122, 95, 68, 145, 82, 89, 64, 21, 30, 71, 52, 165, 94, 101, 72, 183, 246, 203, 140, 271, 154, 161, 112, 97

Offset: 0

Antti Karttunen, Mar 01 2016

Permutation of natural numbers obtained from the Lucky sieve. Note the indexing: Domain starts from 0, range from 1.
This sequence can be represented as a binary tree. After a(1)=2, each left hand child is obtained by applying A269369 to the parent, and each right hand child is obtained by doubling the contents of the parent node, when the parent node contains n:
                                    1
                                    |
                 ...................2...................
                3                                       4
      7......../ \........6                   5......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         11       12         19       10         17       16
13 18   35  28     39  22   29  24     27  38   23  20     61  34   41  32
etc.
Sequence A269375 is obtained from the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A269378.
Cf. A269369.
Cf. A000959 (with 2 inserted between 1 and 3 forms the left edge of the tree).
Related permutation: A269375.
Cf. also A252753, A269387.

a(0) = 1, a(1) = 2; after which, a(2n) = A269369(a(n)), a(2n+1) = 2*a(n).
As a composition of related permutations:
a(n) = A260742(A269387(n)).

A269384 Permutation of natural numbers: a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 18, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 35, 30, 25, 38, 39, 48, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 58, 57, 86, 87, 108, 69, 104, 59, 54, 49, 74, 75, 94, 77, 116, 95, 84, 41, 62, 63, 78, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 112, 105, 158, 159, 198, 133, 200

Offset: 1

Antti Karttunen, Mar 01 2016

Permutation obtained from the Ludic sieve.
This sequence can be represented as a binary tree. For n > 2, each left hand child is obtained by doubling the contents of the parent node and subtracting one, and each right hand child is obtained by applying A269382(n), when the parent node contains n:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       18         13       20         11       10
17 26   27  34     29  44   35  30     25  38   39  48     21  32   19  12
etc.

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

Inverse: A269383.
Cf. A000035, A001511, A003602, A255127, A269382.
Cf. also A269385, A269387 and also A249814, A269374.

a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.
a(1) = 1, a(2n) = A269382(a(n)), a(2n+1) = (2*a(n+1))-1.
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

cp's OEIS Frontend

A269379 a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).

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A269171 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269379(a(A268674(2n+1))).

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A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269380(n)).

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A302025 Permutation of natural numbers mapping ordinary factorization to "Ludic factorization": a(1) = 1, a(2n) = 2*a(n), a(A003961(n)) = A269379(a(n)).

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A269385 Tree of Ludic sieve, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A269379(a(n)).

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A269377 Tree of Lucky sieve: a(0) = 1, a(1) = 2; after which a(2n) = A269369(a(n)), a(2n+1) = 2*a(n).

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A269384 Permutation of natural numbers: a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.

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A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269380(n)).