A243344 -id:A243344 - cp's OEIS frontend

A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2a(k).

1, 3, 7, 2, 15, 5, 31, 6, 14, 11, 63, 4, 13, 29, 23, 30, 127, 10, 9, 62, 27, 59, 47, 12, 28, 61, 22, 126, 255, 21, 19, 8, 125, 55, 119, 26, 95, 25, 57, 58, 123, 45, 253, 46, 60, 511, 43, 254, 20, 18, 39, 124, 17, 54, 251, 118, 111, 239, 53, 94, 191, 51, 24, 56

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This is an instance of an "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A005117/A013929 (numbers which are squarefree/not squarefree) is entangled with complementary pair odd/even numbers (A005408/A005843).
Thus this shares with permutation A243352 the property that each term of A005117 is mapped bijectively to a unique odd number and likewise each term of A013929 is mapped (bijectively) to a unique even number. However, instead of placing terms into those positions in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself.
Are there any other fixed points than 1, 13, 54, 120, 1389, 3183, ... ?

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

Inverse: A243344.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A243352 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1; thereafter, if A008966(n) = 0 (i.e., n is a term of A013929, not squarefree), a(n) = 2*a(A057627(n)); otherwise a(n) = 2*a(A013928(n+1)-1)+1 (where A057627 and A013928(n+1) give the number of integers <= n divisible/not divisible by a square greater than one).

For all n, A000035(a(n)) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) modulo 2. The same property holds for A243352.

A245606 Permutation of natural numbers: a(1) = 1, a(2n) = 1 + A003961(a(n)), a(2n+1) = A003961(1+a(n)). [Where A003961(n) shifts the prime factorization of n one step left].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 15, 16, 11, 26, 21, 22, 13, 12, 27, 28, 25, 36, 81, 82, 19, 14, 45, 52, 125, 56, 39, 40, 29, 18, 33, 46, 17, 126, 99, 100, 31, 50, 51, 226, 41, 626, 129, 130, 89, 24, 63, 34, 35, 176, 87, 154, 59, 344, 825, 298, 115, 86, 189, 190, 43, 32, 105, 76, 23, 66, 57, 88, 53, 20

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

The even bisection halved gives A245608. The odd bisection incremented by one and halved gives A245708.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245605.
Cf. A003961, A243501, A064216, A245608, A245706, A245708.
Similar entanglement permutations: A244319, A156552, A243071, A243288, A243344, A243346, A244322, A245604, A245614, A227413, A237126.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A245606(n) = if(1==n,1,if(0==(n%2),1+A003961(A245606(n/2)),A003961(1+A245606((n-1)/2))))

;; With memoization-macro definec.
(definec (A245606 n) (cond ((= 1 n) 1) ((even? n) (A243501 (A245606 (/ n 2)))) (else (A003961 (+ 1 (A245606 (/ (- n 1) 2)))))))

a(1) = 1, a(2n) = A243501(a(n)), a(2n+1) = A003961(1+a(n)).
As a composition of related permutations:
a(n) = A064216(A245608(n)).

A243346 a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)), where A005117 are squarefree and A013929 are nonsquarefree numbers.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 12, 5, 9, 13, 24, 10, 18, 19, 32, 7, 16, 14, 25, 21, 36, 38, 63, 15, 27, 30, 49, 31, 50, 53, 84, 11, 20, 26, 45, 22, 40, 39, 64, 34, 54, 59, 96, 62, 99, 103, 162, 23, 44, 42, 72, 47, 80, 79, 126, 51, 81, 82, 128, 86, 136, 138, 220, 17, 28, 33, 52, 41, 68, 73, 120

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This permutation entangles complementary pair A005843/A005408 (even/odd numbers) with complementary pair A005117/A013929 (numbers which are squarefree/are not squarefree).

Antti Karttunen, Table of n, a(n) for n = 1..8189
Index entries for sequences that are permutations of the natural numbers

Inverse: A243345.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A075378, A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)).

For all n > 1, A008966(a(n)) = A000035(n+1), or equally, mu(a(n)) + 1 = n modulo 2, where mu is Moebius mu (A008683). [A property shared with a simpler variant A075378].

A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 7, 10, 18, 24, 17, 64, 13, 14, 33, 20, 36, 48, 11, 19, 34, 25, 65, 128, 26, 28, 15, 66, 40, 72, 21, 96, 22, 38, 37, 68, 50, 130, 49, 35, 256, 52, 129, 27, 29, 56, 67, 30, 41, 132, 73, 80, 144, 42, 97, 192, 44, 23, 39, 76, 74, 136, 69, 100

Offset: 1

Antti Karttunen, Jun 03 2014

Any other fixed points than 1, 2, 6, 9, 135, 147, 914, ... ?

Any other points than 4, 21, 39, 839, 4893, 12884, ... where a(n) = n-1 ?

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

Inverse: A243346.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, and for n>1, if mu(n) = 0, a(n) = 1 + 2*a(A057627(n)), otherwise a(n) = 2*a(A013928(n)), where mu is Moebius mu function (A008683).

For all n > 1, A000035(a(n)+1) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) + 1 modulo 2.

A244322 Permutation of natural numbers: a(1)=1, a(2n) = A244991(a(n)), a(2n+1) = A244990(1+a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 13, 10, 12, 15, 14, 16, 18, 17, 19, 22, 24, 25, 27, 20, 21, 23, 26, 31, 29, 30, 28, 32, 35, 34, 37, 33, 36, 40, 38, 45, 43, 47, 49, 50, 52, 55, 54, 41, 39, 44, 42, 46, 48, 51, 53, 64, 61, 60, 57, 62, 58, 59, 56, 66, 63, 69, 71, 68, 70, 75

Offset: 1

Antti Karttunen, Jul 22 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A244321.
Cf. A244990, A244991, A244992.
Similar entanglement permutations: A227413, A237126, A243288, A243344, A243346.

a(1)=1, a(2n) = A244991(a(n)), a(2n+1) = A244990(1+a(n)).

For all n >= 1, A244992(a(n)) = 1 - A000035(n).

A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

Original entry on oeis.org

1, 4, 2, 8, 3, 9, 5, 12, 6, 16, 7, 18, 10, 20, 11, 24, 13, 25, 14, 27, 15, 28, 17, 32, 19, 36, 21, 40, 22, 44, 23, 45, 26, 48, 29, 49, 30, 50, 31, 52, 33, 54, 34, 56, 35, 60, 37, 63, 38, 64, 39, 68, 41, 72, 42, 75, 43, 76, 46, 80, 47, 81, 51, 84, 53, 88, 55, 90, 57, 92, 58, 96

Offset: 1

Amarnath Murthy, Oct 16 2003

nonn

From Antti Karttunen, Jun 04 2014: (Start)
Squarefree (A005117) and nonsquarefree numbers (A013929) interleaved, the former at odd n and the latter at even n.
A243344 is a a "recursivized" variant of this permutation. Like this one, it also satisfies the given simple identity linking the parity of n with the Moebius mu-function. (End)

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers.

Inverse: A243352.
Bisections: A005117, A013929.
Similar permutations: A075378, A073846, A237056, A072061, A094077, A167151, A088609, A243344.
Cf. A000035, A008966.

With[{max = 100}, s = Select[Range[max], SquareFreeQ]; ns = Complement[Range[max], s]; Riffle[s[[1 ;; Length[ns]]], ns]] (* Amiram Eldar, Mar 04 2024 *)

(define (A088610 n) (if (even? n) (A013929 (/ n 2)) (A005117 (/ (+ 1 n) 2))))

From Antti Karttunen, Jun 04 2014: (Start)
a(2n) = A013929(n), a(2n-1) = A005117(n).
For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). (End)

More terms from Ray Chandler, Oct 18 2003

A245614 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = A026424(a(k)), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = A028260(1+a(k)).

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 10, 5, 9, 12, 11, 16, 15, 24, 18, 8, 17, 14, 22, 20, 26, 19, 29, 25, 28, 36, 35, 55, 39, 44, 31, 13, 30, 27, 21, 38, 34, 51, 46, 42, 37, 57, 40, 47, 32, 52, 45, 62, 56, 50, 68, 60, 82, 81, 67, 121, 86, 93, 105, 72, 65, 79, 33, 59, 64, 23, 53, 48, 41, 58, 49, 85, 71, 77, 66, 111, 99

Offset: 1

Antti Karttunen, Jul 27 2014

This shares with the permutation A122111 the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245613.
Cf. A026424, A028260, A066829, A244988, A244989, A244990, A244991, A244992, A122111.
Similar permutations: A244321, A244322, A245604, A227413, A237126, A243288, A243344, A243346, A245606.

a(1) = 1, and for n > 1, if A244992(n) = 1, a(n) = A026424(a(A244989(n))), otherwise a(n) = A028260(1+a(A244988(n)-1)).
As a composition of related permutations:
a(n) = A245604(A244321(n)).
For all n >= 1, A244992(n) = A066829(a(n)).

A285112 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 8, 7, 25, 14, 27, 10, 12, 13, 16, 11, 49, 39, 125, 22, 28, 42, 81, 15, 20, 19, 18, 21, 169, 26, 32, 17, 121, 79, 343, 65, 117, 205, 625, 35, 44, 43, 56, 69, 84, 133, 243, 23, 45, 33, 40, 31, 361, 30, 24, 34, 63, 277, 2197, 41, 52, 53, 64, 29, 289, 199, 1331, 130, 6241, 563, 2401, 106, 325, 193, 351, 335, 1025, 1030, 3125, 58

Offset: 0

Antti Karttunen, Apr 17 2017

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A005117(1+n), and each right hand child as A065642(n), when the parent node contains n >= 2:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........9                   6......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       25         14       27         10       12         13       16
11 49   39  125    22  28   42  81     15  20   19  18     21  169  26  32
etc.

Antti Karttunen, Table of n, a(n) for n = 0..116
Index entries for sequences that are permutations of the natural numbers

Inverse: A285111.
Cf. A005117, A065642.
Similar or related permutations: A243344, A243346, A252753, A277696, A284572.
Cf. also arrays A284457 & A284311.

a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

A284572 Permutation of natural numbers: a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(1+a(n)).

Original entry on oeis.org

1, 2, 4, 3, 9, 6, 25, 5, 8, 14, 20, 10, 49, 39, 52, 7, 12, 13, 27, 22, 45, 33, 63, 15, 121, 79, 80, 65, 50, 85, 2809, 11, 16, 19, 169, 21, 28, 42, 56, 35, 529, 73, 92, 55, 68, 103, 128, 23, 32, 199, 244, 130, 100, 131, 243, 106, 132, 82, 153, 139, 172, 4619, 5620, 17, 18, 26, 289, 31, 40, 277, 340, 34, 44, 43, 841, 69, 1849, 91, 171, 58, 48

Offset: 1

Antti Karttunen, Apr 17 2017

This sequence can be represented as a binary tree. Each left hand child is produced as A005117(1+n), and each right hand child as A065642(1+n), when the parent node contains n:
                                    1
                 ................../ \..................
                2                                       4
      3......../ \........9                   6......../ \........25
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  5       8          14       20         10       49         39       52
7  12   13 27      22  45   33  63     15  121  79  80     65  50   85 2809
etc.
Compare to A285112.

Antti Karttunen, Table of n, a(n) for n = 1..166
Index entries for sequences that are permutations of the natural numbers

Inverse: A284571.
Cf. A005117, A065642.
Similar or related permutations: A243344, A243346, A277696, A285112.

a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(1+a(n)).

cp's OEIS Frontend

A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2*a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2*a(k).

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A245606 Permutation of natural numbers: a(1) = 1, a(2n) = 1 + A003961(a(n)), a(2n+1) = A003961(1+a(n)). [Where A003961(n) shifts the prime factorization of n one step left].

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A243346 a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)), where A005117 are squarefree and A013929 are nonsquarefree numbers.

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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2*a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2*a(k)+1.

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A244322 Permutation of natural numbers: a(1)=1, a(2n) = A244991(a(n)), a(2n+1) = A244990(1+a(n)).

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A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

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A245614 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = A026424(a(k)), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = A028260(1+a(k)).

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A285112 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

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A284572 Permutation of natural numbers: a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(1+a(n)).

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A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2a(k).

A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.