A257727 -id:A257727 - cp's OEIS frontend

A257728 Permutation of natural numbers: a(1)=1; a(2n) = not_an_oddprime(1+a(n)), a(2n+1) = oddprime(a(n)).

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 13, 10, 17, 12, 19, 14, 23, 18, 37, 15, 29, 21, 43, 16, 31, 26, 61, 20, 41, 28, 71, 22, 47, 34, 89, 27, 67, 52, 163, 24, 53, 42, 113, 32, 79, 60, 193, 25, 59, 45, 131, 38, 103, 84, 293, 30, 73, 57, 181, 40, 109, 95, 359, 33, 83, 65, 223, 49, 149, 119, 463, 39, 107, 91, 337, 72, 241, 209, 971, 35, 97, 74, 251, 58

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).
This sequence can be represented as a binary tree. Each left hand child is produced as A065090(1+n), and each right hand child as A065091(n), when a parent contains n >= 1:
                                    |
                 ...................1...................
                2                                       3
      4......../ \........5                   6......../ \........7
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  8       11          9       13         10       17         12       19
14 23   18  37      15 29   21  43     16  31   26  61     20  41   28  71
etc.
Because all odd primes are odd, it means that even terms can only occur in even positions (together with odd composites, A071904, 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 = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A257727.
Cf. A002808, A065090, A065091, A071904.
Related or similar permutations: A246377, A246378, A257726, A257729, A257802.
Differs from A255004 for the first time at n=17, where a(17) = 23, while A255004(17) = 15.

A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A257728(n) = if(n<3, n, if(!(n%2), A002808(A257728(n/2)-1), prime(1+A257728((n-1)/2))));
for(n=1, 4096, write("b257728.txt", n, " ", A257728(n)));

;; With memoizing definec-macro.
(definec (A257728 n) (cond ((< n 2) n) ((even? n) (A065090 (+ 1 (A257728 (/ n 2))))) (else (A065091 (A257728 (/ (- n 1) 2))))))

a(1) = 1; a(2n) = A065090(1+a(n)), a(2n+1) = A065091(a(n)).
As a composition of other permutations:
a(n) = A257729(A246378(n)).
a(n) = A257802(A257726(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)).

A257801 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = lucky(1+a(n)), a(not_an_oddprime(n)) = unlucky(a(n-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 6, 11, 8, 13, 14, 25, 10, 17, 12, 15, 19, 33, 20, 35, 16, 21, 24, 18, 22, 27, 45, 43, 28, 31, 47, 23, 29, 34, 26, 51, 30, 38, 59, 63, 57, 115, 39, 42, 61, 37, 32, 40, 46, 36, 66, 73, 41, 52, 78, 83, 76, 49, 146, 67, 53, 56, 81, 50, 44, 79, 54, 60, 48, 163, 86, 87, 95, 55, 68, 101, 107, 171, 98, 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: A257802.
Cf. A000959, A000720, A010051, A050505, A062298, A065090, A065091.
Related or similar permutations: A257726, A257727, A257730, A257731.

a(1) = 1; a(2) = 2; if A010051(n) = 1 [i.e., when n is an (odd) prime] then a(n) = A000959(1+a(A000720(n)-1)), otherwise a(n) = A050505(a(A062298(n))).
As a composition of other permutations:
a(n) = A257726(A257727(n)).
a(n) = A257731(A257730(n)).

cp's OEIS Frontend

A257728 Permutation of natural numbers: a(1)=1; a(2n) = not_an_oddprime(1+a(n)), a(2n+1) = oddprime(a(n)).

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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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A257801 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = lucky(1+a(n)), a(not_an_oddprime(n)) = unlucky(a(n-1)).

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