A257801 -id:A257801 - cp's OEIS frontend

A257726 a(0)=0; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n)+1), where lucky = A000959, unlucky = A050505.

0, 1, 2, 3, 4, 7, 5, 9, 6, 13, 11, 25, 8, 15, 14, 33, 10, 21, 19, 51, 17, 43, 35, 115, 12, 31, 22, 67, 20, 63, 45, 163, 16, 37, 29, 93, 27, 79, 66, 273, 24, 73, 57, 223, 47, 171, 146, 723, 18, 49, 42, 151, 30, 99, 88, 385, 28, 87, 83, 349, 59, 235, 203, 1093, 23, 69, 50, 193, 40, 135, 119, 559, 38, 129, 102, 475, 86, 367, 335, 1983, 34, 111

Offset: 0

Antti Karttunen, May 06 2015

This sequence can be represented as a binary tree. Each left hand child is produced as A050505(n), and each right hand child as A000959(1+n), when a parent contains n >= 1:
                                    0
                                    |
                 ...................1...................
                2                                       3
      4......../ \........7                   5......../ \........9
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  6       13         11       25          8       15         14       33
10 21   19  51     17  43   35  115     12 31   22  67     20  63   45  163
etc.
Because all lucky numbers are odd, it means that even terms can only occur in even positions (together with odd unlucky numbers, for each one of which there is a separate infinite cycle), while terms in odd positions are all odd.

Antti Karttunen, Table of n, a(n) for n = 0..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A257725.
Cf. A000959, A050505, A005408, A005843.
Related or similar permutations: A237126, A246378, A257728, A257731, A257733, A257801.
Cf. also A183089 (another similar permutation, but with a slightly different definition, resulting the first differing term at n=9, where a(9) = 13, while A183089(9) = 21).
Cf. also A257735 - A257738.

a(0)=0; after which, a(2n) = A050505(a(n)), a(2n+1) = A000959(a(n)+1).
As a composition of other permutations. For all n >= 1:
a(n) = A257731(A246378(n)).
a(n) = A257733(A237126(n)).
a(n) = A257801(A257728(n)).

A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 3, 16, 14, 12, 23, 8, 17, 26, 24, 21, 13, 35, 5, 15, 27, 39, 53, 36, 33, 22, 51, 10, 43, 25, 37, 40, 56, 75, 52, 49, 83, 34, 72, 18, 19, 62, 59, 38, 54, 57, 101, 78, 102, 74, 69, 114, 89, 50, 98, 28, 30, 86, 73, 82, 41, 55, 76, 80, 134, 106, 149, 135, 100, 94, 11, 150, 47, 120, 70, 130, 42, 45, 103, 117, 99, 112, 167, 58, 77

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here composite(n) = n-th composite = A002808(n), prime(n) = n-th prime = A000040(n), oddprime(n) = n-th odd prime = A065091(n) = A000040(n+1), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

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

Inverse: A257729.
Cf. A000040, A000720, A002808, A010051, A062298, A065090, A065091.
Related or similar permutations: A246378, A257727, A257732, A257801, A236854.

a(1) = 1; if A000035(n) = 1 and A010051(n) = 1 [i.e., when n is an odd prime], then a(n) = A000040(a(A000720(n)-1)), otherwise a(n) = A002808(a(A062298(n))). [Here A062298(n) gives the index of n among numbers larger than 1 which are not odd primes, 1 for 2, 2 for 4, 3 for 6, etc.]
As a composition of other permutations:
a(n) = A246378(A257727(n)).
a(n) = A257732(A257801(n)).

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 9, 14, 11, 16, 20, 24, 13, 18, 15, 28, 22, 32, 17, 40, 48, 26, 36, 30, 21, 56, 25, 44, 64, 34, 80, 96, 19, 52, 72, 60, 29, 42, 23, 112, 50, 88, 33, 128, 68, 160, 192, 38, 41, 104, 144, 120, 58, 84, 49, 46, 27, 224, 100, 176, 66, 256, 37, 136, 320, 384, 31, 76, 57, 82, 208, 288, 240, 116, 45

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here oddprime(n) = n-th odd prime = A065091(n) = A000040(n+1), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

			For n=2, which is the second natural number >= 1 that is not an odd prime [2 = A065090(2)], we compute 2*a(1) = 2 = a(2).
For n=4, which is A065090(3), we compute 2*a(3-1) = 2*2 = 4.
For n=5, and 5 is the second odd prime [5 = A065091(2)], thus a(5) = 1 + 2*a(2) = 5.
For n=9, which is the sixth natural number >= 1 not an odd prime (9 = A065090(6)), we compute 2*a(6-1) = 2*5 = 10.
For n=11, which is the fourth odd prime [11 = A065091(4)], we compute 1 + 2*a(4) = 1 + 2*4 = 9, thus a(11) = 9.

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

Inverse: A257728.
Cf. A000720, A010051, A062298, A065090, A065091.
Related or similar permutations: A246377, A246378, A257725, A257730, A257801.

a(1) = 1; a(2) = 2; and for n > 2, if A010051(n) = 1 [i.e., when n is a prime], then a(n) = 1 + 2*a(A000720(n)-1), otherwise a(n) = 2*a(A062298(n)).
As a composition of other permutations:
a(n) = A246377(A257730(n)).
a(n) = A257725(A257801(n)).

A257802 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = oddprime(a(n-1)), a(unlucky(n)) = not_an_oddprime(1+a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 10, 7, 14, 9, 16, 11, 12, 17, 22, 15, 25, 18, 20, 23, 26, 33, 24, 13, 36, 27, 30, 34, 38, 31, 48, 19, 35, 21, 51, 47, 39, 44, 49, 54, 45, 29, 66, 28, 50, 32, 70, 59, 65, 37, 55, 62, 68, 75, 63, 42, 90, 40, 69, 46, 94, 41, 81, 88, 52, 61, 76, 83, 85, 92, 100, 53, 86, 101, 58, 120, 56, 67, 93, 64

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here lucky(n) = n-th lucky number = A000959(n), unlucky(n) = n-th unlucky number = A050505(n), oddprime(n) = n-th odd prime = A065091(n), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

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

Inverse: A257801.
Cf. A000959, A050505, A065090, A065091, A109497, A145649.
Related or similar permutations: A257725, A257728, A257729, A257732.

a(1) = 1; if A145649(n) = 1 [i.e., when n is lucky] then a(n) = A065091(a(A109497(n)-1)), otherwise a(n) = A065090(1+a(n-A109497(n))).
As a composition of other permutations:
a(n) = A257728(A257725(n)).
a(n) = A257729(A257732(n)).

cp's OEIS Frontend

A257726 a(0)=0; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n)+1), where lucky = A000959, unlucky = A050505.

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A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

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A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).

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A257802 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = oddprime(a(n-1)), a(unlucky(n)) = not_an_oddprime(1+a(n)).

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cp's OEIS Frontend

A257726 a(0)=0; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n)+1), where lucky = A000959, unlucky = A050505.

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Formula

A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

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Formula

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2*a(n), a(not_an_oddprime(n)) = 2*a(n-1).

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Formula

A257802 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = oddprime(a(n-1)), a(unlucky(n)) = not_an_oddprime(1+a(n)).

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A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).