A269171 -id:A269171 - cp's OEIS frontend

A269393 Permutation of natural numbers: a(n) = A269171(3*n) / 3.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 38, 39, 40, 41, 42, 43, 44, 49, 46, 47, 48, 37, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 73, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 81, 90, 91, 76, 77, 78, 79, 80, 85

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

The first non-fixed term is a(37)=45.

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

Inverse: A269394.

Cf. A269171, A269395.

(define (A269393 n) (/ (A269171 (* 3 n)) 3))

a(n) = A269171(3*n) / 3.

A269395 Permutation of natural numbers: a(n) = A269171(A269393(n)) = A269171(A269171(3*n)/3).

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, 45, 46, 39, 40, 43, 42, 47, 44, 31, 50, 53, 48, 41, 38, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 89, 74, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 81, 90, 85, 92, 59, 78, 91, 80, 79

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Composition of A269171 with a permutation of natural numbers obtained from its trisection.

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

Inverse: A269396.
Cf. A269393.
Differs from A255407 and A269171 for the first time at n=37, which here a(37)=45, instead of 41.

(define (A269395 n) (A269171 (A269393 n)))

a(n) = A269171(A269393(n)) = A269171(A269171(3*n)/3).

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

Original entry on oeis.org

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.

A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(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, 25, 20, 21, 22, 19, 24, 23, 26, 27, 28, 29, 30, 49, 32, 33, 34, 35, 36, 31, 50, 39, 40, 37, 42, 41, 44, 45, 38, 43, 48, 55, 46, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 98, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 62, 75, 100, 67, 78, 85, 80, 81

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2244
Index entries for sequences that are permutations of the natural numbers

Inverse: A269171.
Cf. A000035, A003309, A008578, A083221, A250469, A260738, A260739, A269380.
Related or similar permutations: A260741, A260742, A269356, A269358, A255422.
Cf. also A269394 (a(3n)/3) and A269396.
Differs from A255408 for the first time at n=38, where a(38) = 50, while A255408(38) = 38.

a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, A250469(a(A269380(n))).
a(1) = 1, for n > 1, a(n) = A083221(A260738(n), a(A260739(n))).
As a composition of other permutations:
a(n) = A252755(A269386(n)).
a(n) = A252753(A269388(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]

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)).

A269387 Tree of Ludic sieve: a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 19, 18, 21, 16, 11, 14, 27, 20, 35, 30, 33, 24, 31, 38, 51, 36, 49, 42, 45, 32, 13, 22, 39, 28, 65, 54, 57, 40, 59, 70, 87, 60, 79, 66, 69, 48, 55, 62, 111, 76, 125, 102, 105, 72, 85, 98, 123, 84, 109, 90, 93, 64, 17, 26, 63, 44, 95, 78, 81, 56, 113, 130, 159, 108, 139, 114, 117, 80

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 applying A269379 to the parent's contents, and each right hand child is obtained by doubling the parent's contents:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........6                   9......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       10         15       12         19       18         21       16
11 14   27  20     35  30   33  24     31  38   51  36     49  42   45  32
etc.
Sequence A269385 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: A269388.
Cf. A003309 (left edge of the tree).
Cf. A269379.
Related permutations: A260741, A269171, A269385.
Cf. also A252753, A269377.

a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n).
As a composition of other permutations:
a(n) = A269171(A252753(n)).
a(n) = A260741(A269377(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.]

A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 9, 18, 17, 20, 31, 22, 25, 24, 21, 26, 27, 28, 19, 30, 29, 32, 49, 34, 71, 36, 37, 38, 39, 40, 41, 42, 43, 44, 107, 46, 47, 48, 119, 50, 51, 52, 35, 54, 89, 56, 101, 58, 53, 60, 61, 62, 63, 64, 115, 66, 67, 68, 173, 70, 55, 72, 33, 74, 75, 76, 131, 78, 77, 80, 167

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

			For n=9 we first find what number is below 9 in square array A083221, which is 25, then we find what number is above 25 in square array A255127, which is 23, thus a(9) = 23.

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

Inverse: A269356.
Cf. A250469, A269380.
Cf. also arrays A083221 & A255127.
More recursed variant: A269357. Cf. also permutations A266645, A255407, A269171.

(define (A269355 n) (A269380 (A250469 n)))

a(n) = A269380(A250469(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes the even numbers.]

A269358 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A268674(A269379(2n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 17, 10, 11, 12, 13, 14, 15, 16, 19, 34, 29, 20, 25, 22, 9, 24, 23, 26, 27, 28, 31, 30, 21, 32, 73, 38, 53, 68, 37, 58, 39, 40, 41, 50, 43, 44, 107, 18, 47, 48, 33, 46, 51, 52, 59, 54, 71, 56, 137, 62, 101, 60, 61, 42, 63, 64, 109, 146, 67, 76, 121, 106, 35, 136, 97, 74, 75, 116, 79, 78, 131, 80, 197

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

This is a variant of A269356, from which it differs for the first time at n=18.

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

Inverse: A269357.

Cf. A000035, A268674, A269379, A269356, A269171.

a(1) = 1, after which, for even n, a(n) = 2*a(n/2) and for odd n, a(n) = A269356(n) = A268674(A269379(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

cp's OEIS Frontend

A269393 Permutation of natural numbers: a(n) = A269171(3*n) / 3.

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A269395 Permutation of natural numbers: a(n) = A269171(A269393(n)) = A269171(A269171(3*n)/3).

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

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A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

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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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A269387 Tree of Ludic sieve: a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*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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A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

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A269358 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A268674(A269379(2n+1)).

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