A250470 -id:A250470 - cp's OEIS frontend

A252756 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A250470(2n+1)).

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 255, 28, 9, 62, 511, 24, 11, 126, 29, 60, 1023, 26, 2047, 16, 25, 254, 27, 20, 4095, 510, 61, 56, 8191, 18, 16383, 124, 17, 1022, 32767, 48, 23, 22, 21, 252, 65535, 58, 19, 120, 57, 2046, 131071, 52, 262143, 4094, 125, 32, 59, 50, 524287, 508, 49, 54, 1048575, 40

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

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

Inverse: A252755.
Similar permutations: A243071, A252754, A054429, A250246.
Cf. also A250470, A253556 - A253559.
Differs from A243071 for the first time at n=21, where a(21) = 9, while A243071(21) = 29.

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A250470(2n+1)).
As a composition of related permutations:
a(n) = A054429(A252754(n)).
a(n) = A243071(A250246(n)).

A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 6, 13, 10, 11, 20, 9, 14, 71, 22, 7, 26, 19, 12, 23, 16, 21, 24, 41, 18, 53, 28, 31, 56, 29, 38, 107, 58, 67, 74, 61, 32, 197, 40, 25, 68, 59, 50, 137, 64, 73, 62, 49, 44, 227, 76, 27, 80, 55, 30, 89, 34, 43, 66, 37, 48, 91, 46, 69, 60, 35, 42, 65, 70, 15, 78, 47, 36, 119, 52

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..6560
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A250469, A250470, A263273.
Cf. A265369, A265904, A266190, A266401 (other conjugates or similar derivations of A263273).
Cf. also A250471, A250472, A264996, A266404, A266415, A266416, A266645, A266646.

(define (A266403 n) (A250470 (A263273 (A250469 n))))

a(n) = A250470(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266415(A266645(n)) = A266646(A266416(n)).
a(n) = A250472(A264996(A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 8, 19, 9, 10, 23, 29, 12, 15, 31, 14, 37, 41, 16, 43, 25, 18, 47, 21, 20, 53, 59, 22, 27, 61, 24, 67, 71, 26, 35, 73, 28, 79, 33, 30, 83, 55, 32, 39, 89, 34, 97, 101, 36, 103, 107, 38, 109, 45, 40, 65, 49, 42, 51, 113, 44, 127, 85, 46, 131, 137, 48, 77, 57, 50, 139, 149, 52, 63, 151, 54, 95, 157, 56, 163, 121, 58, 167, 69, 60

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For n > 1, a(n) tells which number is located immediately above n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains 2n - 1.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A250471.
Odd bisection of A250470. The other bisection: A250479.
Cf. A055396, A064216, A078898, A083140, A083221, A114881, A114883.

a(1) = 1, a(n) = A083221(A055396(2*n - 1)-1, A078898(2*n - 1)).

a(n) = A250470(2*n - 1).

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.

Original entry on oeis.org

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).

A266646 Permutation of natural numbers: a(n) = A250470(A003961(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 16, 13, 12, 15, 28, 17, 26, 19, 22, 21, 14, 23, 46, 25, 18, 51, 34, 29, 36, 31, 82, 27, 20, 35, 76, 37, 24, 33, 64, 41, 56, 43, 40, 69, 30, 47, 136, 49, 50, 39, 52, 53, 126, 55, 100, 45, 32, 59, 106, 61, 38, 111, 244, 65, 66, 67, 58, 57, 78, 71, 226, 73, 42, 99, 70, 77, 86, 79, 190, 249, 44, 83, 166, 85

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..767
Index entries for sequences that are permutations of the natural numbers

Inverse: A266645.
Cf. A000035, A020639, A055396, A003961, A250470.
Related permutations: A249817, A249818, A266403, A266415.

(define (A266646 n) (A250470 (A003961 n)))

a(n) = A250470(A003961(n)).
As a composition of related permutations:
a(n) = A266403(A266415(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(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).]

A266415 Permutation of natural numbers: a(n) = A250470(A263273(A003961(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 10, 13, 6, 11, 22, 9, 20, 71, 28, 7, 18, 19, 16, 23, 14, 21, 64, 41, 26, 227, 58, 31, 74, 29, 82, 53, 12, 67, 52, 61, 24, 107, 46, 25, 30, 59, 40, 65, 56, 73, 190, 49, 44, 197, 76, 27, 230, 55, 172, 137, 38, 43, 220, 37, 32, 571, 244, 69, 60, 35, 34, 89, 72, 15, 154, 47, 68, 479, 70

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..767
Index entries for sequences that are permutations of the natural numbers

Inverse: A266416.
Cf. A000035, A003961, A250470, A263273.
Related permutations: A048673, A250472, A264985, A264996, A266403, A266646.

(define (A266415 n) (A250470 (A263273 (A003961 n))))

a(n) = A250470(A263273(A003961(n))).
As a composition of related permutations:
a(n) = A266403(A266646(n)).
a(n) = A250472(A264996(A048673(n))) = A250472(1+A264985(-1+A048673(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266404 Self-inverse permutation of natural numbers: a(n) = A250470(A030101(A250469(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 17, 10, 7, 12, 13, 20, 25, 16, 9, 18, 23, 14, 53, 22, 19, 28, 15, 36, 27, 24, 29, 40, 37, 32, 33, 34, 83, 26, 31, 42, 51, 30, 47, 38, 59, 44, 101, 76, 41, 60, 73, 68, 39, 52, 21, 84, 107, 56, 131, 72, 43, 48, 89, 80, 125, 64, 65, 66, 109, 50, 99, 82, 71, 58, 49, 74, 151, 46, 239, 78, 97, 62, 173, 70, 35, 54

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A030101, A057889, A250469, A250470.
Cf. A265329, A266402 (other conjugates or similar derivations of A057889).
Cf. also A266403.

(define (A266404 n) (A250470 (A030101 (A250469 n))))

a(n) = A250470(A030101(A250469(n))) = A250470(A057889(A250469(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A250479 Even bisection of A250470: a(n) = A250470(2n).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 27, 5, 34, 23, 53, 6, 59, 29, 20, 1, 39, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16, 37, 79, 10, 63, 41, 46, 7, 83, 12, 65, 19, 58, 43, 75, 2, 89, 25, 28, 9, 97, 26, 101

Offset: 1

Antti Karttunen, Dec 08 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A250470, A250472 (the other bisection).

Differs from A064989 for the first time at n=55, where a(55) = 27, while A064989(55) = 21.

Scheme
```
(define (A250479 n) (A250470 (+ n n)))
```

a(n) = A250470(2*n).

cp's OEIS Frontend

A252756 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A250470(2n+1)).

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A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

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A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

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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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A266646 Permutation of natural numbers: a(n) = A250470(A003961(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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A266415 Permutation of natural numbers: a(n) = A250470(A263273(A003961(n))).

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A266404 Self-inverse permutation of natural numbers: a(n) = A250470(A030101(A250469(n))).

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A250479 Even bisection of A250470: a(n) = A250470(2n).

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A252756 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A250470(2n+1)).