A252753 -id:A252753 - cp's OEIS frontend

A253555 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A250470(2n+1)); also binary width of terms of A252754 and A252756.

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 4, 6, 9, 5, 4, 7, 5, 6, 10, 5, 11, 5, 5, 8, 5, 5, 12, 9, 6, 6, 13, 5, 14, 7, 5, 10, 15, 6, 5, 5, 5, 8, 16, 6, 5, 7, 6, 11, 17, 6, 18, 12, 7, 6, 6, 6, 19, 9, 6, 6, 20, 6, 21, 13, 8, 10, 6, 7, 22, 7, 7, 14, 23, 6, 6, 15, 6, 8, 24, 6, 6, 11, 6, 16, 7, 7, 25, 6, 9, 6

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

a(n) tells how many iterations of A253554 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A252753 and A252755.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000040, A000079, A001248, A253554.
Cf. also A252753, A252754, A252755, A252756, A253557, A253558, A253559.
Differs from A252464 for the first time at n=21, where a(21) = 4, while A252463(21) = 5.

a(1) = 0; for n > 1: a(n) = 1 + a(A253554(n)).
a(n) = A029837(1+A252754(n)) = A029837(1+A252756(n)).
a(n) = A253556(n) + A253557(n).
Other identities.
For all n >= 1:
a(A000079(n)) = n. [I.e., a(2^n) = n.]
a(A000040(n)) = n.
a(A001248(n)) = n+1.
For n >= 2, a(n) = A253558(n) + A253559(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)).

A253554 a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 4, 5, 7, 6, 11, 7, 6, 8, 13, 9, 17, 10, 8, 11, 19, 12, 9, 13, 10, 14, 23, 15, 29, 16, 12, 17, 15, 18, 31, 19, 14, 20, 37, 21, 41, 22, 16, 23, 43, 24, 25, 25, 18, 26, 47, 27, 21, 28, 20, 29, 53, 30, 59, 31, 22, 32, 27, 33, 61, 34, 24, 35, 67, 36, 71, 37, 26, 38, 35, 39, 73, 40, 28, 41, 79, 42, 33, 43, 30

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Divide the even numbers by two, and for odd numbers n >= 3, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1.

For any number n >= 2 in binary trees A252753 and A252755, a(n) gives the number which is the parent of n.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Bisections: A000027 and A250472.
Cf. A253555 (the number of iterations needed to reach 1 from n).
Cf. A055396, A078898, A250470, A252753, A252755.
Differs from A252463 for the first time at n=21, where a(21) = 8, while A252463(21) = 10.

(define (A253554 n) (cond ((<= n 1) n) ((even? n) (/ n 2)) (else (A250470 n))))

a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

A253556 a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 5, 3, 2, 0, 6, 1, 7, 2, 1, 4, 8, 1, 2, 5, 3, 3, 9, 2, 10, 0, 2, 6, 3, 1, 11, 7, 4, 2, 12, 1, 13, 4, 1, 8, 14, 1, 3, 2, 2, 5, 15, 3, 2, 3, 3, 9, 16, 2, 17, 10, 5, 0, 4, 2, 18, 6, 2, 3, 19, 1, 20, 11, 6, 7, 4, 4, 21, 2, 4, 12, 22, 1, 3, 13, 3, 4, 23, 1, 3, 8, 1, 14, 5, 1, 24, 3, 7, 2, 25

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Consider the binary tree illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located at), a(n) gives the number of odd numbers > 1 encountered on the path (i.e., excluding the final 1 from the count but including the starting n if it was odd).

Antti Karttunen, Table of n, a(n) for n = 1..8192

One less than A253558.
Powers of two, A000079, gives the positions of zeros.
Cf. A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253557, A253559.
Differs from A252735 for the first time at n=21, where a(21) = 1, while A252735(21) = 3.

a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).
a(n) = A253555(n) - A253557(n).
a(n) = A253558(n) - 1.
a(n) = A080791(A252754(n)). [Number of nonleading 0-bits in A252754(n).]
Other identities. For all n >= 2:
a(n) = A000120(A252756(n)) - 1. [One less than the binary weight of A252756(n).]

A253559 a(1) = 0; for n>1: a(n) = A253557(n) - 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 1, 1, 1, 2, 0, 2, 0, 4, 2, 1, 1, 3, 0, 1, 1, 3, 0, 3, 0, 2, 3, 1, 0, 4, 1, 2, 2, 2, 0, 2, 2, 3, 2, 1, 0, 3, 0, 1, 1, 5, 1, 3, 0, 2, 3, 2, 0, 4, 0, 1, 1, 2, 1, 2, 0, 4, 2, 1, 0, 4, 2, 1, 2, 3, 0, 4, 2, 2, 4, 1, 1, 5, 0, 2, 1, 3, 0, 3, 0, 3, 3, 1, 0, 3, 0, 3, 1, 4, 0, 3, 3, 2, 3, 1, 1, 4, 1, 1, 3, 2, 2, 2, 0, 6

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located), a(n) gives the number of even numbers > 2 encountered on the path (i.e., excluding the 2 from the count but including the starting n if it was even).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Essentially, one less than A253557.
A008578 gives the positions of zeros.
Cf. A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253556, A253558.
Differs from A252736 for the first time at n=21, where a(21) = 2, while A252736(21) = 1.

(define (A253559 n) (if (= 1 n) 0 (+ -1 (A253557 n))))

a(n) = A080791(A252756(n)). [Number of nonleading 0-bits in A252756(n).]
a(1) = 0; for n>1: a(n) = A253557(n) - 1.
Other identities. For all n >= 2:
a(n) = A000120(A252754(n)) - 1. [One less than the binary weight of A252754(n).]
a(n) = A253555(n) - A253558(n).

A253558 a(n) = A253556(n) + 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 3, 1, 7, 2, 8, 3, 2, 5, 9, 2, 3, 6, 4, 4, 10, 3, 11, 1, 3, 7, 4, 2, 12, 8, 5, 3, 13, 2, 14, 5, 2, 9, 15, 2, 4, 3, 3, 6, 16, 4, 3, 4, 4, 10, 17, 3, 18, 11, 6, 1, 5, 3, 19, 7, 3, 4, 20, 2, 21, 12, 7, 8, 5, 5, 22, 3, 5, 13, 23, 2, 4, 14, 4, 5, 24, 2, 4, 9, 2, 15, 6, 2, 25, 4, 8, 3, 26, 3

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located at), a(n) gives the number of odd numbers encountered on the path (i.e., including both the final 1 and the starting n if it was odd).

Antti Karttunen, Table of n, a(n) for n = 1..8192

One more than A253556.
Powers of two, A000079, gives the positions of ones.
Cf. A000040, A253555, A253557, A253559.
After n=1, differs from A061395 for the first time at n=21, where a(21) = 2, while A061395(21) = 4.

Scheme
```
(define (A253558 n) (+ 1 (A253556 n)))
```

a(n) = A253556(n) + 1.
a(n) = A080791(A252754(n)) + 1. [One more than the number of nonleading 0-bits in A252754(n).]
Other identities.
For all n >= 1:
a(A000040(n)) = n.
For all n >= 2:
a(n) = A000120(A252756(n)). [Binary weight of A252756(n).]
a(n) = A253555(n) - A253559(n).

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)).

A285112 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 8, 7, 25, 14, 27, 10, 12, 13, 16, 11, 49, 39, 125, 22, 28, 42, 81, 15, 20, 19, 18, 21, 169, 26, 32, 17, 121, 79, 343, 65, 117, 205, 625, 35, 44, 43, 56, 69, 84, 133, 243, 23, 45, 33, 40, 31, 361, 30, 24, 34, 63, 277, 2197, 41, 52, 53, 64, 29, 289, 199, 1331, 130, 6241, 563, 2401, 106, 325, 193, 351, 335, 1025, 1030, 3125, 58

Offset: 0

Antti Karttunen, Apr 17 2017

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A005117(1+n), and each right hand child as A065642(n), when the parent node contains n >= 2:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........9                   6......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       25         14       27         10       12         13       16
11 49   39  125    22  28   42  81     15  20   19  18     21  169  26  32
etc.

Antti Karttunen, Table of n, a(n) for n = 0..116
Index entries for sequences that are permutations of the natural numbers

Inverse: A285111.
Cf. A005117, A065642.
Similar or related permutations: A243344, A243346, A252753, A277696, A284572.
Cf. also arrays A284457 & A284311.

a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

A279348 a(1) = 1, for n > 1, if A079559(n) = 0, a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 9, 10, 15, 11, 8, 12, 14, 25, 27, 18, 35, 13, 20, 30, 21, 33, 22, 39, 49, 16, 24, 28, 50, 65, 51, 54, 77, 17, 36, 70, 57, 87, 26, 55, 85, 40, 60, 42, 63, 95, 66, 121, 45, 44, 78, 69, 81, 98, 147, 119, 32, 48, 56, 100, 130, 125, 159, 102, 143, 19, 108, 154, 105, 207, 34, 145, 215, 72, 140, 114, 75, 91, 174, 133, 117, 52

Offset: 1

Antti Karttunen, Dec 12 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: A279349.
Cf. A005187, A055938, A079559, A250469, A256992.
Related or similar permutations: A250245, A252753, A252755, A279338, A279341, A279343.

(definec (A279348 n) (cond ((= 1 n) n) ((zero? (A079559 n)) (* 2 (A279348 (A256992 n)))) (else (A250469 (A279348 (A256992 n))))))

a(1) = 1, for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).
As a composition of other permutations:
a(n) = A250245(A279338(n)).
a(n) = A252753(A279343(n)).
a(n) = A252755(A279341(n)).

cp's OEIS Frontend

A253555 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A250470(2n+1)); also binary width of terms of A252754 and A252756.

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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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A253554 a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

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A253556 a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).

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A253559 a(1) = 0; for n>1: a(n) = A253557(n) - 1.

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A253558 a(n) = A253556(n) + 1.

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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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A285112 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

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A279348 a(1) = 1, for n > 1, if A079559(n) = 0, a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).

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