A257731 -id:A257731 - 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)).

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)).

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)).

A257729 Permutation of natural numbers: a(1)=1; a(prime(n)) = oddprime(a(n)), a(composite(n)) = not_an_oddprime(1+a(n)).

Original entry on oeis.org

1, 3, 7, 2, 19, 6, 5, 12, 4, 28, 71, 10, 17, 9, 20, 8, 13, 40, 41, 95, 16, 26, 11, 15, 30, 14, 21, 56, 109, 57, 359, 125, 25, 38, 18, 24, 31, 44, 22, 32, 61, 77, 29, 143, 78, 445, 73, 162, 36, 54, 27, 35, 23, 45, 62, 33, 46, 84, 43, 104, 179, 42, 185, 105, 545, 98, 181, 208, 51, 75, 503, 39, 59, 50, 34, 63, 85, 48, 103, 64, 114, 60, 37

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).

			As an initial value we have a(1) = 1.
2 is the first prime (= A000040(1)), so we take the a(1)-th odd prime, A065091(1) = 3, thus a(2) = 3.
3 is the second prime, thus we take a(2)-th odd prime, A065091(3) = 7, thus a(3) = 7.
4 is the first composite, thus we take a(1)-th number larger than one which is not an odd prime, and that is A065090(1+1) = 2, thus a(4) = 2.
5 is the third prime, thus we take a(3)-th odd prime, which is A065091(7) = 19, thus a(5) = 19.

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

Inverse: A257730.
Cf. A000040, A002808, A000720, A010051, A065090, A065091, A065855.
Related or similar permutations: A257728, A246377, A257731, A257802, A236854.

A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n};
A257729(n) = { if(1==n,n,if(isprime(n),prime(1+A257729(primepi(n))),if(4==n,2,A002808(A257729(n-primepi(n)-1)-1))))};
for(n=1, 10000, write("b257729.txt", n, " ", A257729(n)));
\\ After M. F. Hasler's PARI-code for A236854.

;; With memoizing definec-macro.
(definec (A257729 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A065091 (A257729 (A000720 n)))) (else (A065090 (+ 1 (A257729 (A065855 n)))))))
;; Alternatively, by composing other permutations:
(define (A257729 n) (A257728 (A246377 n)))

a(1) = 1; if A010051(n) = 1 [i.e., if n is a prime], then a(n) = A065091(a(A000720(n))), otherwise a(n) = A065090(1+a(A065855(n))).
As a composition of other permutations:
a(n) = A257728(A246377(n)).
a(n) = A257802(A257731(n)).

A257733 Permutation of natural numbers: a(1) = 1, a(ludic(n)) = lucky(1+a(n-1)), a(nonludic(n)) = unlucky(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 3, 9, 2, 33, 5, 7, 14, 4, 45, 163, 8, 15, 11, 20, 6, 25, 59, 203, 12, 22, 17, 63, 28, 13, 10, 35, 78, 235, 251, 18, 30, 24, 83, 39, 19, 1093, 16, 47, 101, 31, 290, 67, 309, 26, 41, 43, 34, 107, 53, 27, 1283, 87, 23, 61, 128, 42, 354, 88, 376, 21, 36, 55, 57, 46, 137, 115, 70, 38, 1499, 321, 112, 32, 81, 161, 56, 1401, 430, 113, 454, 29, 48, 49

Offset: 1

Antti Karttunen, May 06 2015

Antti Karttunen, Table of n, a(n) for n = 1..8587
Index entries for sequences that are permutations of the natural numbers

Inverse: A257734.
Cf. A000959, A050505, A003309, A192607, A192490, A192512, A236863.
Related or similar permutations: A237427, A255422, A257726, A257731.
Cf. also A256486, A256487.
Differs from A257731 for the first time at n=19, where a(19) = 203, while A257731(19) = 63.

a(1) = 1; for n > 1: if A192490(n) = 1 [i.e., if n is ludic], then a(n) = A000959(1+a(A192512(n)-1)), otherwise a(n) = A050505(a(A236863(n))).
As a composition of other permutations:
a(n) = A257731(A255422(n)).
a(n) = A257726(A237427(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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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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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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A257729 Permutation of natural numbers: a(1)=1; a(prime(n)) = oddprime(a(n)), a(composite(n)) = not_an_oddprime(1+a(n)).

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A257733 Permutation of natural numbers: a(1) = 1, a(ludic(n)) = lucky(1+a(n-1)), a(nonludic(n)) = unlucky(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and lucky = A000959, unlucky = A050505.

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