A257725 -id:A257725 - cp's OEIS frontend

A257735 Permutation of natural numbers: a(n) = A257725(A183089(n)).

1, 2, 3, 4, 5, 6, 7, 8, 17, 10, 9, 12, 25, 14, 13, 16, 57, 34, 33, 20, 49, 18, 11, 24, 105, 50, 19, 28, 29, 26, 15, 32, 153, 114, 113, 68, 385, 66, 65, 40, 769, 98, 97, 36, 81, 22, 21, 48, 609, 210, 209, 100, 133, 38, 37, 56, 1537, 58, 35, 52, 257, 30, 27, 64, 1953, 306, 163, 228, 1409, 226, 225, 136, 13313, 770, 99

Offset: 1

Antti Karttunen, May 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A257736.

Cf. A183089, A257725, A257737.

(define (A257735 n) (A257725 (A183089 n)))

a(n) = A257725(A183089(n)).

A257737 Permutation of natural numbers: a(n) = A183089(A257725(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 14, 31, 16, 17, 18, 29, 20, 87, 42, 23, 24, 13, 26, 40, 28, 112, 56, 141, 32, 15, 34, 19, 36, 517, 54, 39, 143, 74, 177, 49, 44, 22, 46, 27, 48, 925, 618, 37, 71, 53, 179, 96, 220, 64, 58, 30, 60, 38, 62, 63, 1088, 737, 50, 51, 92, 4129, 70, 222, 122, 259, 271, 8037, 84, 77, 41

Offset: 1

Antti Karttunen, May 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..230
Index entries for sequences that are permutations of the natural numbers

Inverse: A257738.

Cf. A183089, A257725, A257735.

(define (A257737 n) (A183089 (A257725 n)))

a(n) = A183089(A257725(n)).

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

Original entry on oeis.org

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)).

A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

Original entry on oeis.org

1, 4, 2, 9, 6, 16, 7, 12, 3, 26, 14, 21, 23, 8, 13, 39, 24, 33, 35, 15, 53, 22, 56, 36, 17, 49, 51, 25, 75, 34, 37, 78, 5, 52, 27, 69, 101, 72, 38, 102, 50, 54, 43, 106, 10, 74, 40, 94, 73, 134, 83, 98, 55, 135, 70, 76, 62, 141, 18, 100, 57, 125, 19, 99, 175, 114, 41, 130, 167, 77, 176, 95, 89, 104, 137, 86, 184, 28, 149, 133, 80, 164, 30

Offset: 1

Antti Karttunen, May 06 2015

In other words, a(1) = 1 and for n > 1, if n is the k-th lucky number larger than 1 [i.e., n = A000959(k+1)] then a(n) = nthprime(a(k)), otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], then a(n) = nthcomposite(a(k)).

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

Inverse: A257731.
Cf. A000040, A000959, A002808, A050505, A109497, A145649.
Related or similar permutations: A246378, A255422, A257725, A257734.
Cf. also A032600, A255553, A255554.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = A000040(a(A109497(n)-1)), otherwise a(n) = A002808(a(n-A109497(n))).
As a composition of other permutations:
a(n) = A246378(A257725(n)).
a(n) = A255422(A257734(n)).

A257690 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = (2a(n))-1, a(unlucky(n)) = 2n, where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 12, 7, 16, 10, 24, 11, 14, 15, 32, 20, 48, 22, 28, 9, 30, 64, 40, 23, 96, 44, 56, 18, 60, 13, 128, 31, 80, 46, 192, 19, 88, 112, 36, 120, 26, 47, 256, 62, 160, 92, 384, 21, 38, 27, 176, 224, 72, 240, 52, 94, 512, 124, 320, 184, 768, 29, 42, 76, 54, 63, 352, 39, 448, 144, 480, 95, 104, 43, 188, 1024, 248, 55

Offset: 1

Antti Karttunen, May 09 2015

nonn

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

Inverse permutation: A183089.
Cf. A005408, A005843, A000959, A050505, A109497, A145649.
Cf. also A257725 (similar permutation with a slightly different definition, resulting the first differing term at n=13, where a(13) = 11, while A257725(13) = 9).
Cf. also A257735 - A257738.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = (2*a(A109497(n)))-1, otherwise a(n) = 2*a(n-A109497(n)). [Where A109497(n) gives the number of lucky numbers <= 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)).

A257734 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = ludic(1+a(n-1)), a(unlucky(n)) = nonludic(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and ludic = A003309, nonludic = A192607.

Original entry on oeis.org

1, 4, 2, 9, 6, 16, 7, 12, 3, 26, 14, 20, 25, 8, 13, 38, 22, 31, 36, 15, 61, 21, 54, 33, 17, 45, 51, 24, 81, 32, 41, 73, 5, 48, 27, 62, 119, 69, 35, 105, 46, 57, 47, 96, 10, 65, 39, 82, 83, 151, 115, 92, 50, 135, 63, 76, 64, 124, 18, 86, 55, 106, 23, 108, 189, 146, 43, 118, 193, 68, 169, 84, 91, 100, 149, 85, 156, 28, 179, 111, 74, 136, 34

Offset: 1

Antti Karttunen, May 06 2015

nonn

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

Inverse: A257733.
Cf. A000959, A050505, A003309, A109497, A145649, A192607.
Related or similar permutations: A237126, A255421, A257725, A257732.
Cf. also A256486, A256487.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = A003309(1+a(A109497(n)-1)), otherwise a(n) = A192607(a(n-A109497(n))).
As a composition of other permutations:
a(n) = A255421(A257732(n)).
a(n) = A237126(A257725(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

A257735 Permutation of natural numbers: a(n) = A257725(A183089(n)).

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A257737 Permutation of natural numbers: a(n) = A183089(A257725(n)).

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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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A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

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A257690 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = (2*a(n))-1, a(unlucky(n)) = 2*n, where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

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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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A257734 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = ludic(1+a(n-1)), a(unlucky(n)) = nonludic(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and ludic = A003309, nonludic = A192607.

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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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A257690 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = (2a(n))-1, a(unlucky(n)) = 2n, where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

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