A237126 -id:A237126 - cp's OEIS frontend

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.

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.

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)).

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).

A245702 Permutation of natural numbers: a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)), where A014580(n) = binary code for n-th irreducible polynomial over GF(2) and A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 11, 8, 7, 6, 13, 9, 47, 17, 31, 14, 25, 12, 19, 10, 59, 20, 37, 15, 319, 62, 87, 24, 185, 42, 61, 21, 137, 34, 55, 18, 97, 27, 41, 16, 415, 76, 103, 28, 229, 49, 67, 22, 3053, 373, 433, 79, 647, 108, 131, 33, 1627, 222, 283, 54, 425, 78, 109, 29, 1123, 166, 203, 45, 379, 71, 91, 26, 731, 121, 145, 36, 253, 53, 73, 23

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Inverse: A245701.
Cf. A000035, A014580, A091242, A091225.
Similar entanglement permutations: A193231, A227413, A237126, A243288, A245703, A245704.

allocatemem(123456789);
a014580 = vector(2^18);
a091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; a014580[i] = n, j++; a091242[j] = n); n++)
A245702(n) = if(1==n, 1, if(0==(n%2), a014580[A245702(n/2)], a091242[A245702((n-1)/2)]));
for(n=1, 383, write("b245702.txt", n, " ", A245702(n)));

;; With memoizing definec-macro.
(definec (A245702 n) (cond ((< n 2) n) ((even? n) (A014580 (A245702 (/ n 2)))) (else (A091242 (A245702 (/ (- n 1) 2))))))

a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)).
As a composition of related permutations:
a(n) = A245703(A227413(n)).
Other identities:
For all n >= 1, 1 - A091225(a(n)) = A000035(n). [Maps even numbers to binary representations of irreducible GF(2) polynomials (= A014580) and odd numbers to the corresponding representations of reducible polynomials].

A246202 Permutation of natural numbers: a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)), where A091242(n) = binary code for n-th reducible polynomial over GF(2) and A014580(n) = binary code for n-th irreducible polynomial over GF(2).

Original entry on oeis.org

1, 4, 2, 8, 11, 5, 3, 14, 31, 17, 47, 9, 13, 6, 7, 21, 61, 42, 185, 24, 87, 62, 319, 15, 37, 20, 59, 10, 19, 12, 25, 29, 109, 78, 425, 54, 283, 222, 1627, 33, 131, 108, 647, 79, 433, 373, 3053, 22, 67, 49, 229, 28, 103, 76, 415, 16, 41, 27, 97, 18, 55, 34, 137, 39, 167, 134, 859, 98, 563, 494, 4225, 70, 375, 331, 2705, 264, 2011, 1832, 19891, 44

Offset: 1

Antti Karttunen, Aug 19 2014

This sequence can be represented as a binary tree. Each left hand child is produced as A091242(n), and each right hand child as A014580(n), when the parent contains n:
                                     |
                  ...................1...................
                 4                                       2
       8......../ \.......11                   5......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  14       31         17       47          9       13          6       7
21  61   42  185    24  87   62  319     15 37   20  59      10 19   12 25
etc.
Because 2 is the only even term in A014580, it implies that, apart from a(3)=2, all other odd positions contain an odd number. There are also odd numbers in the even bisection, precisely all the terms of A246156 in some order, together with all even numbers larger than 2 that are also there. See also comments in A246201.

Inverse: A246201.
Similar or related permutations: A245702, A246162, A246164, A246204, A237126, A003188, A054429, A193231, A260422, A260426.
Cf. A014580, A091242, A091225, A246156.

allocatemem((2^31)+(2^30));
uplim = (2^25) + (2^24);
v014580 = vector(2^24);
v091242 = vector(uplim);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < uplim), if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++)
A246202(n) = if(1==n, 1, if(0==(n%2), v091242[A246202(n/2)], v014580[A246202((n-1)/2)]));
for(n=1, 638, write("b246202.txt", n, " ", A246202(n)));
\\ Works with PARI Version 2.7.4. - Antti Karttunen, Jul 25 2015
(Scheme, with memoization-macro definec)
(definec (A246202 n) (cond ((< n 2) n) ((odd? n) (A014580 (A246202 (/ (- n 1) 2)))) (else (A091242 (A246202 (/ n 2))))))

a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)).
As a composition of related permutations:
a(n) = A245702(A054429(n)).
a(n) = A246162(A003188(n)).
a(n) = A193231(A246204(n)).
a(n) = A246164(A193231(n)).
a(n) = A260426(A260422(n)).
Other identities:
For all n > 1, A091225(a(n)) = A000035(n). [After 1, maps even numbers to binary representations of reducible GF(2) polynomials and odd numbers to the corresponding representations of irreducible polynomials, in some order. A246204 has the same property].

A255421 Permutation of natural numbers: a(1) = 1, a(p_n) = ludic(1+a(n)), a(c_n) = nonludic(a(n)), where p_n = n-th prime, c_n = n-th composite number and ludic = A003309, nonludic = A192607.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 19, 20, 21, 25, 22, 24, 26, 27, 28, 29, 34, 37, 30, 31, 32, 36, 33, 41, 35, 38, 39, 43, 40, 47, 42, 49, 52, 53, 44, 45, 46, 51, 48, 61, 57, 50, 54, 55, 59, 67, 56, 71, 64, 58, 66, 70, 72, 97, 60, 62, 63, 77, 69, 83, 65, 81

Offset: 1

Antti Karttunen, Feb 23 2015

nonn

This can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case a complementary pair ludic/nonludic numbers (A003309/A192607) is intertwined with a complementary pair prime/composite numbers (A000040/A002808).

			When n = 19 = A000040(8) [the eighth prime], we look for the value of a(8), which is 8 [all terms less than 19 are fixed because the beginnings of A003309 and A008578 coincide up to A003309(8) = A008578(8) = 17], and then take the eighth ludic number larger than 1, which is A003309(1+8) = 23, thus a(19) = 23.
When n = 20 = A002808(11) [the eleventh composite], we look for the value of a(11), which is 11 [all terms less than 19 are fixed, see above], and then take the eleventh nonludic number, which is A192607(11) = 19, thus a(20) = 19.
When n = 30 = A002808(19) [the 19th composite], we look for the value of a(19), which is 23 [see above], and then take the 23rd nonludic number, which is A192607(23) = 34, thus a(30) = 34.

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

Inverse: A255422.
Cf. A000040, A000720, A002808, A003309, A007097, A008578, A065855, A010051, A192607, A255420, A255324.
Related or similar permutations: A237126, A246377, A245703, A245704, A255407, A255408.

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003309(1+a(A000720(n))), otherwise a(n) = A192607(a(A065855(n))).
As a composition of other permutations:
a(n) = A237126(A246377(n)).
Other identities.
a(A007097(n)) = A255420(n). [Maps iterates of primes to the iterates of Ludic numbers.]

A235491 Self-inverse permutation of natural numbers: complementary pair ludic/nonludic numbers (A003309/A192607) entangled with the same pair in the opposite order, nonludic/ludic. See Formula.

Original entry on oeis.org

0, 1, 4, 9, 2, 16, 7, 6, 25, 3, 61, 26, 17, 14, 13, 115, 5, 12, 359, 119, 67, 47, 43, 36, 791, 8, 11, 41, 3017, 81, 811, 407, 247, 227, 179, 7525, 23, 38, 37, 221, 34015, 27, 503, 22, 7765, 3509, 1943, 21, 1777, 1333, 93625, 97, 193, 146, 181, 1717, 486721, 121, 4493, 91, 96839, 10, 40217, 20813, 89

Offset: 0

Antti Karttunen, Feb 07 2014

nonn

The permutation is self-inverse (an involution), meaning that a(a(n)) = n for all n.

			For n=2, with 2 being the second ludic number (= A003309(4)), the value is computed as nonludic(a(2-1)) = nonludic(a(1)) = 4, the first nonludic number, thus a(2) = 4.
For n=5, with 5 being the fourth ludic number (= A003309(4)), the value is computed as nonludic(a(4-1)) = nonludic(a(3)) = nonludic(9) = 16, thus a(5) = 16.
For n=6, with 6 being the second nonludic number (= A192607(2)), the value is computed as ludic(a(2)+1) = ludic(4+1) = ludic(5) = 7, thus a(6) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..74 (The terms after a(32) were computed with the help of 100000 term b-file uploaded by Donovan Johnson for A003309.)
Index entries for sequences that are permutations of the natural numbers

Cf. A236854 (a similar permutation constructed from prime and composite numbers).
Cf. A237126/A237427 (entanglement permutations between ludic/nonludic <-> odd/even numbers).
Cf. also A003309, A192607, A192490, A192512, A236863.

a(0)=0, a(1)=1, and for n > 1, if n is k-th ludic number (i.e., n = A003309(k)), then a(n) = nonludic(a(k-1)); otherwise, when n is k-th nonludic number (i.e., n = A192607(k)), then a(n) = ludic(a(k)+1), where ludic numbers are given by A003309, and nonludic numbers by A192607.

a(0)=0, a(1)=1, and for n > 1, if A192490(n)=1 (n is ludic) a(n) = A192607(a(A192512(n)-1)); otherwise (n is nonludic), a(n) = A003309(1+(a(A236863(n)))).

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)).

A266418 Permutation of natural numbers: a(1) = 1, a(2n) = A266410(a(n)), a(2n+1) = A266409(1+a(n)).

Original entry on oeis.org

1, 4, 2, 10, 6, 7, 3, 22, 20, 15, 11, 16, 12, 9, 5, 40, 53, 37, 45, 29, 33, 24, 21, 31, 35, 25, 23, 19, 18, 13, 8, 68, 111, 85, 156, 64, 104, 75, 123, 51, 74, 56, 87, 43, 59, 39, 48, 54, 80, 61, 90, 46, 60, 42, 57, 36, 44, 34, 41, 27, 26, 17, 14, 107, 210, 167, 387, 133, 276, 229, 573, 101, 198, 158, 351, 120

Offset: 1

Antti Karttunen, Jan 28 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A266410(n), and each right hand child as A266409(1+n), when the parent node contains n:
                                     |
                  ...................1...................
                 4                                       2
      10......../ \........6                   7......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  22       20         15       11         16       12          9       5
40  53   37  45     29  33   24  21     31  35   25  23      19 18   13 8
etc.

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A266417.
Cf. A266409, A266410.
Similar or related permutations: A237126, A266638.

a(1) = 1, after which: a(2n) = A266410(a(n)), a(2n+1) = A266409(1+a(n)).
As a composition of related permutations:
a(n) = A266638(A237126(n)).

A252758 Permutation of natural numbers: a(1) = 1, a(2n) = A251726(a(n)), a(2n+1) = A251727(a(n)).

Original entry on oeis.org

1, 2, 10, 3, 14, 12, 38, 4, 20, 17, 44, 15, 40, 61, 92, 5, 22, 25, 57, 21, 51, 72, 102, 18, 46, 64, 94, 108, 132, 191, 182, 6, 26, 29, 60, 35, 68, 101, 124, 27, 58, 85, 116, 135, 152, 221, 198, 23, 52, 75, 106, 115, 138, 193, 184, 239, 206, 311, 242, 499, 333, 467, 318, 7, 28, 36, 69, 43, 76, 107, 130, 54, 87, 127, 145, 217, 196, 283, 231, 37

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..959
Index entries for sequences that are permutations of the natural numbers

Inverse: A252757.
Similar permutations: A243288, A227413, A237126.
Cf. A251726, A251727.

a(1) = 1, a(2n) = A251726(a(n)), a(2n+1) = A251727(a(n)).

cp's OEIS Frontend

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

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A245702 Permutation of natural numbers: a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)), where A014580(n) = binary code for n-th irreducible polynomial over GF(2) and A091242(n) = binary code for n-th reducible polynomial over GF(2).

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A246202 Permutation of natural numbers: a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)), where A091242(n) = binary code for n-th reducible polynomial over GF(2) and A014580(n) = binary code for n-th irreducible polynomial over GF(2).

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A255421 Permutation of natural numbers: a(1) = 1, a(p_n) = ludic(1+a(n)), a(c_n) = nonludic(a(n)), where p_n = n-th prime, c_n = n-th composite number and ludic = A003309, nonludic = A192607.

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A235491 Self-inverse permutation of natural numbers: complementary pair ludic/nonludic numbers (A003309/A192607) entangled with the same pair in the opposite order, nonludic/ludic. See Formula.

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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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A266418 Permutation of natural numbers: a(1) = 1, a(2n) = A266410(a(n)), a(2n+1) = A266409(1+a(n)).

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A252758 Permutation of natural numbers: a(1) = 1, a(2n) = A251726(a(n)), a(2n+1) = A251727(a(n)).

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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.